Tuesday, August 18, 2020

Number Facts for every Year Date, 271- 300

The 271st Day of the Year
271 is the 58th prime number, the larger of a pair of twin primes, and is the sum of eleven consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43).

271 is also the difference of two consecutive cubes, 103 - 93. Such prime numbers are called Cuban Primes, and were named by the British mathematician Allan Joseph Champneys Cunningham in 1923.

Using the English alphabet code, a = 1, b = 2, etc, there are exactly 271 positive numbers that give larger numbers when you write out their English names and add the letters *primecurios

271 is the first three digits of e=2.718281828459045....

Prime Curios states that the largest prime factor of any five digit repdigit, is 271.

271^2 = 73441, and its reversal, 14437 is prime, and 271^3 = 19902511, and its reversal, 1150991 is also prime. *Prime Curios And now you wonder???

271 is the smallest number between numbers with cubes for factors. 270 = 2 x 3^3 x 5, and 272 = 2^4 x 17

Of all the primes between 2 and 269, none of them has period or five for their reciprocal. 1/271 = .00369..., which explains why all five digit repdigits are divisible by 271.

271 is a 4n+3 prime, so it is not the sum of two squares. It is however, the difference of two consecutive squares, 136^2 - 135^2. Of the six permutations of the digits of 271, four of them are centered hexagonal number. Since they must always be odd, you can quickly figure out which two are omitted.

271 = 15+16^2 (n + (n+1)^2). Counting 0+1^2, it is the 16th such number.

The 272nd Day of the Year
272 = 24·17, and is the sum of four consecutive primes (61 + 67 + 71 + 73).

272 is also a Pronic or Heteromecic number, the product of two consecutive factors, 16x17 (which makes it twice a triangular #).

And 272 is a palindrome, and the sum of its digits, 11, is also a palindrome. (can you find the next?)

272 = 16^2 + 4^2 and the difference of two squares 69^2 - 67^2 and also 36^2 - 32^2 , and 21^2 - 13^2
And speaking of that last one, a fun Fibonacci fact, the difference of the squares of consecutive Fibonacci numbers, is the product of the one before times the one after, 8 x 34 = 272.  This is true for all differences of consecutive Fibonacci numbers greater than one.  

272 = 24·17, sum of four consecutive primes (61 + 67 + 71 + 73)

There are 272 ways to partition 39 into prime parts.

272 can be written as the sum of distinct powers of 4, 272 = 4^4 + 4^2,

All the digits of 272 are primes, and the sum of the digits is prime
272 is divisible by four, and so 272 = 69^2 - 67^2. It is also divisible by eight, and so it is = 36^2 - 32^2, and it is also divisible by 16 so, 21^2 - 13^2
The 273rd Day of the Year
273oK(to the nearest integer)is the freezing point of water, or 0oC

OOOOH wait, 273 = 13*7*3, and 1373 is also prime.. and There are only two sphenic numbers consisting of concatenation of distinct prime numbers, this is the smaller of the two.(sphenic or wedge numbers are products of three distinct primes) *Prime curios

273 is a repdigit in hexdecimal, or base 16 (111) 16^2 + 16 + 1, and in base 20(vigesimal), where it looks like a bad report card (DD)

273 = 47^2 - 44^2 and also 137^2 - 136^2

Prime Curios says that 273^10 - 10^273 is the smallest n for which this expression is prime. There are no more year days that exhibit this relationship.

Prime Curios includes this tasty factoid, the prime factors of 273 = 3 x 7 x 13, and if concatenated in reverse order, 1373, it is prime.

273 is a Palindrome in base 2(100010001). 2^8 + 2^4 + 2^0. Also a Palindrome in base 4 (10101) , 4^4 + 4^2 + 4^0.

273 is the product of two consecutive Fibonacci numbers (13 x 21) and is thus a Golden rectangle Number. Each of these as they gown larger approximate a Golden Rectangle. 21/13=1.615....


The 274th Day of the Year
274 is a tribonacci number.The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are 0, 0, 1, 1, 2, 4, 7,..Are there more Tribonacci number in the year Days?

274 is also the sum  of five cubes, 23 + 2 3 + 23 + 53 + 53, and of three triangular numbers 78 + 91 + 105. They are consecutive triangular numbers because their differences are consecutive numbers, 91-78 = 13 and 105 - 91 = 14. In 1796, Gauss proved that every positive integer could be expressed as the sum of three triangular numbers.

274 is an example of Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity) 274= 2 * 137 and 2+ 7 +4 = 13 = 2 + 1 + 3 + 7. Find another.

Sort of strange one from Prime Curios, "(274 - 2)2 + (274 - 7)7 + (274 - 4)4 is prime." Who would even think to check such things? (Impressive curiosity!)

The centered Triangular numbers start with a single point, and then build a triangle around it with side lengths of 1, then 2, etc. I mention this because 274 is a centered triangular number, The difference between two centered triangular numbers increases by three each time. CT(n)= 3n(n-1)/2 + 1. 274 is the 14th centered triangular number.

137 = 11^2 + 4^2. Can you see how that leads you to 274 = 15^2 + 7^2 because it is twice a number which is the sum of two squares.

The 275th Day of the Year
275 = 138^2 - 137^2 = 30^2 - 25^2

275 is the number of partitions of 28 in which no part occurs only once. (Students might try finding the similar number of partitions for 10, or some smaller number to get a sense for how they grow)

275 is the maximum number. of pieces that can be formed from an annulus with 22 straight lines. 

Imagine taking the number 11 and forming every possible partition of it, and then you count all the parts of all these partitions, you get 275, but a shortcut is to take the largest number in each partition, and add those.... Yep, same number.(shortcuts anyone?). Students might try that with 2,3,4 etc and see if they can figure out why they are the same.

275 is only the fifth number that is expressible as 2^n + 3^n.  275 = 2^5 + 3^5.

The 276th Day of the Year
276 is the sum of twelve consecutive prime numbers (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43).

It's also the sum of three consecutive fifth powers, 1^5 + 2^5 + 3^5 = 276

 And from the trivia file, 276 is the number of rounds of the longest boxing match in history. Jack Jones beat Pat Tunney in a bare-knuckle fight in 1825 after 4 1/2 hours.

If we let S(n) = the sum of the proper divisors of n (so S(8) = 1 + 2 + 4 = 7)  and make a chain S(n1) =  n2 and s(n2) = n3... eventually you get back to a number you already had.  Some numbers  (called perfect numbers) do so immediately. Others form chains of 2 (sociable numbers) or 4 (the most common) or more, but every number will loop back........ almost.  It seems 276 is a maverick and just keeps getting bigger all the time.... No other number is known that does not form a chain.

A Four by Four Magic square using only consecutive prime numbers by A. W. Johnson, Jr.. What's the magic constant??? What day is it?

See 258 for a smaller also using consecutive primes.

276 = 70^2 - 68^2

276 is a palindrome in base 8 (424) and a repdigit in base 22(CC)

And 276 is a triangular number, the sum of the first 23 integers.

The 277th Day of the Year
277 is the 59th prime number, It is also a self number, and is the largest prime self number that can be a day of the year (A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.... The next prime self number is 367, too large to be the number of a day of the year)

The ever-clever Derek Orr pointed out, "Keep going, 277--59th prime, 59--17th prime, 17--7th prime." And "277 = 15 + 15 + 25 + 35 (first four Fibonacci numbers raised to the next Fibonacci number(5) power. What would be next?)"

277 is also a Pythagorean Prime. And as Fermat wrote, all 4n+1 primes are the sum of two squares in only one way. Every prime of the form p = 4n+1 can be the hypotenuse of an integer sided right triangle. In this case the triple is (115, 252, 277) Note all the permutations of two alike one different.

The reciprocals of the 59 primes from 2 to 277, 1/2 + 1/3 + 1/5 + 1/7 + .... + 1/277, is barely larger than 2, the smallest primes. *Prime Curios

The square of 277 is also the sum of two squares (115, 252, 277) (Note that all three numbers are two of one digit of another, and they have all three orders of two alike, one different.

277 = 277 = 139^2 - 138^2

277 is the largest factor of the #7(the product of the first seven primes) + 1. Some write this notation as \( p_7#\), perhaps even clearer, would be P!(7) for prime factorial.

a flat disk can be cut into 277 sections with only 23 straight cuts.

On an infinite chessboard, there are 277 squares a knight can reach in six moves. (I wonder how many he has touched along these 277 trips?)

277 in Roman numerals is sort of Palindromic in that the characters CLVI appear in groups of 21212, CCLXXVII

277 is also a palindrome in base 12, the Duodecimal system. (1B1)

277 = 4^4 + 4^2 + 4^1 + 4^0.

The 278th Day of the Year
278 = 2x2x2x2x2x2x2x2+22

1789 - 278= 1511 is prime (278th prime minus 278).Derek's Daily Math  for more

278 is the fifth smallest number n such that nn starts with the digits of n, \(278^278 =2.78... × 10^679 \)

If you like the game "Brussels Sprouts", then you should know that the total number of moves is always a number that is three more than a multiple of five, like 278> 277 = 14^2 + 9^2 = 139^2 - 138^2

There are 278 primes less than 1800.

The 279th Day of the Year
Every positive integer is the sum of at most 279 eighth powers. See Waring's problem
279 seems to require 24 itself, 2^8 with twenty-three 1^8.

279 = 3^2 + 3^3 + 3^5 (powers are consecutive primes). *Derek Orr It is also 7^3 - 4^3 (PB)
For more fun with daily calendar math see Dereks Daily Math
and Ben Vitale

Another from the prolific Derek Orr, 279 + 10^57 (this means there is a prime gap of more than 279 between this prime and the one before it. 10^57 + 279 is the first prime with 58 digits.)

279 = 1^2 + 2^2 + 7^2 + 15^2, It can not be done with fewer positive squares. (But is that the ONLY sum of four squares?)

279 = 48^2 - 45^2, and also 140^2 - 139^2

279 = \(8!_2 - 7!_2\) This is me persisting against the use of n!! for the alternate factorial, 8 x 6 x 4 x 2.

And Derek Orr has 279 is a divisor of 999,999,999,999,999. If I understand him, it will not divide any shorter string.

The 280th Day of the Year
The 280th day of the year; There are 280 plane trees with ten nodes. As a consequence of this, 18 people around a round table can shake hands without crossing arms in 280 different ways (up to rotations)

The sum of the first 280 consecutive primes mod 280 is prime. *Prime Curios (Stijn Dierckx ‏@Stanny1990 told me there are 108 such days in a year [almost 30% of the days satisfy this property]. Next one in 5 days!)

280 = 71^2 - 69^2 = 37^2 - 33^2 = 4^3 + 6^3

18 people around a round table can shake hands with each other in non-crossing ways, in 280 different ways (this includes rotations) *Wikipedia

280 is a palindrome in base 3(101101)

280+ 1 is prime, and 280^2 + 1 is prime *Derek Orr

\(280 = 10!_3\) = 10 x 7 x 4 x 1

And 280 is a Joy-Giver (Harshad) number, divisible by the sum of its digits.

The 281st Day of the Year
281 is the 60th Prime, and a smaller twin prime and is the sum of the first fourteen prime numbers. It is also the sum of 7 consecutive primes starting at 29

Prime Curios offers that 281 is the larges prime such that the (sum of the factorials from 1 to P ) - 2 is prime

281 is the sixth, and last, day of the year such that the sum of the first k primes is a prime. (I just noticed that all of them except 2, is the smaller of pair of twin primes. Unfortunately, the next one after that is not.)

281 appears in the sequence of primes generated by \(f(x)= x^2 + x + 41 \)  Which is often called the Euler Polynomial. (although Euler actually used  \(x^2 -x + 41\) which is prime for x values from 1 to 40.  Legendre noticed that the positive x form produced the same primes for values from 0 to 39.)

Here are four squares with the same digits 2812 =78961, 2862=81796, 1372=18769,1332 = 17 689.
Since five unique digits can represent 120 different numbers, how many, on average, are squares?

Found this one in the Twitter feed of Jim Wilder@wilderlab in 2016, For day 281, a palindrome: 281=9•8+7•6+5•4+3•2+1+2•3+4•5+6•7+8•9

281 is a prime of the form\(10!_3 + 1\).

If you find 281!, then remove all the Primes (divide by p#281), that value +/- 1 are twin primes.

Another from Prime Curios, There are 281 triangular numbers on the digital clock, including seconds. What's the first one after midnight.

281 is the smallest prime such that the period length of the reciprocal is (p-1)/10.

281 is a Pythagorean Prime, the hypotenuse of (160, 231, 281).

And since 281 is a 4n+1 prime, it is the sum of two squares, 281 = 16^2 + 5^2, and because it is an odd number, 281 = 141^2 - 140^2

The 282nd Day of the Year
there are 282 plane partitions of nine objects. (A plane partition is a two-dimensional array of integers n_(i,j) that are nonincreasing both from left to right and top to bottom and that add up to a given number n.) (That reads much harder than the idea, here is an image of a plane partition of 22 from Mathworld, which, as they say, is worth a thousand words:


282 is the smallest number between twin primes which is a palindrome. Can you find the next?

282 is the largest gap between two successive primes below one billion.

There is no digit you can place after 289 which makes the result a prime. *Derek Orr

The sum of the divisors of 282 is 576, a square number. It is the 15th such number this year, and there will be six more.

And sitting between twin primes, 282 is the average of two consecutive primes.

282 is a palindrome in base 10 (282) and a repdigit in base 9 (333)

282 can be written as the sum of nine positive fifth powers, but it takes twelve positive fourth powers.

An admirable number is a number so that s(n) (the sum of the divisors of n) - 2d (for some divisor d of n)= 2n. For 282 that works with d=6. 576 - 12 = 564 =2 x 282.

The 282nd triangular number is the first triangular number divisible by 47

The 283rd Day of the Year
5 + 81 + 35. and one more way below
It is the 61st Prime, which makes it a prime of prime order. The sum of its digits is also a prime. And it is the concatenation of two primes 2 and 83.*Derek Orr

And 6 x 283 sits between another pair of twin primes. (PB)

283 can be expressed as nn + (n+1)n+1 (Find n.) The largest prime year day for which this is true.

283 = 3^3 + 4^4. *Derek Orr

Like all odd numbers, it is the difference of consecutive squares, 283 = 142^2 - 141^2.

283 = (6! - 5! - 4! - 3! - 2! - 1! - 0!)/2.

283 in base eight is 238, a permutation of its own digits.

283 is a prime of the form 4n+3, Bernard Frénicle de Bessy discovered that such primes cannot be the hypotenuse of a Pythagorean triangle (1676), as opposed to primes of the form 4n+1, which Fermat conjectured always were in 1640.

The smallest amicable pair is (220, 284). 283 is the only prime adjacent to either of them, making it the smallest prime adjacent to an amicable number, and the only such prime that is a year day. There are no primes adjacnt to the next smallest pair.

Prime Cuios says the state of Pennsylvania is 283 miles, west to east.

The 284th Day of the Year
The 284th day of the year; 284 is an amicable (or friendly) number paired with 220. The divisors of 220 add up to 284 and the divisors of 284 add up to 220. Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by Thābit ibn Qurra (826-901).(Can you find the next pair?)

jim wilder ‏@wilderlab pointed out a variation of friendly numbers in degree three... "Along the lines of friendly numbers... 136 = 2³ + 4³ + 4³ and 244 = 1³ + 3³ + 6³"

284 is divisible by four, so it is the difference of two squares, 72^2 - 70^2.

Lagrange's theorem tells us that each positive integer can be written as a sum of four squares. This theorem allows for some squares of zero. If you restrict the set to only those that require four positive squares for the sum is much smaller. 284 is such a number, and is the 46th such number. 289 = 17^2 + 4^2 + 2^2 + 2^2. It seems that there can sometimes be consecutive integers that require four squares (239,240), but never a triplet.

The 51st prime is 231, making 284 the sum of 51 and the 51st prime.

And 284 is a palindrome in base 8(434)

The 285th Day of the Year
285 is a square pyramidal number (like a stack of cannonballs, or oranges with the base in a square)... Or.. the sum of the first nine squares. \( 285 = \sum _{i=1} ^ 9 (i^2) \). 285 is the largest Year day that is a Square Puramidal number.

285 is 555 in base 7.

285 = 3^2 + 5^2 + 7^2 + 9^2 + 11^2

Not sure how rare this is, but just saw it on MAA's Number a Day and was intrigued, 285^2 = 81225 uses the same digits as 135^2 (18225) and 159^2 (25281).

285 is a Joy-giver (Harshad) number, divisible by the sum of its digits.

285 is the difference of two squares, 285 = 31^2-26^2 = 49^2 - 46^2.

A unusual relation to produce the Square Pyramidal numbers, For odd n, (n+n+2+c)^3 - n^3 - (n+2)^3 - (n+4)^3 divided by 524 it gives the nth square Pyramidal number. When n=7, you get 285.

285 is the longer leg of a primative Pythagorean Triple. (68, 285, 293)

The 286th Day of the Year
286 is a sphenic (wedge) number, the product of three distinct primes, 2 x 11 x 13.

If you take the product of each pair of numbers that add up to 12, their sum will be 286. There is a similar pattern for every triangular pyramidal number.

286 is a tetrahedral number (a triangular pyramid, note that 285 was a square pyramidal number, how often can they occur in sequence?) It is the sum of the first eleven triangular numbers, 286 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66

And to top yesterday's curiosity, here are four squares with the same digits 2862=81796, 1372=18769, 1332 = 17 689, 2812 =78961

286 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2 = *Derek Off, It is the last year day which is sum of squaes of first n primes.

286= 7^3 - 7^2 - 7^1 -7^0 *Derek Orr

There are factorials, lesser known subfactorials, and an even more obscure Swinging Factorial. The swinging factorial. The symbol is a backwards s curve. But I'll use n!S. it is given by n! divided by the square of the factorial of the floor function of n/2. The function osscillates up and down after the first few terms. 5!S for instance is is 5!/(2!)^2 = 120 /4 = 30. 6!S is 6!/(3!)^2 = 720/36 = 20 .Explore........

The 287th Day of the Year
287 = 7^3 - 7^2 - 7^1. *Derek Orr

287 is not prime, but it is the sum of three consecutive primes (89 + 97 + 101), and also the sum of five consecutive primes (47 + 53 + 59 + 61 + 67), and wait, it is also the sum of nine consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). *Prime Curios

287 is the smallest non-prime Kynea number and the 4th overall.A Kyneaa number is an integer of the form 4n + 2n+1 − 1, studied by Cletus Emmanuel who named them after a baby girl. The binary expression of these numbers is interesting, 287[2] is 100011111. Each Kynea number has a one, followed by n-1 zeros, followed by n+1 ones. The Keyna primes are all two less than a square number. There are four year days that are Kynea numbers (7, 23, 79, 287), but 287 is the only one that is composite.

287 is the 14th Pentagonal number. Pentagonal numbers are figurate numbers, but unlike the triangular and square numbers, they are not rotationally symmetric. Instead each starts at the same first point and overlaps previous ones. The number in each outer level is a multiple of five, but the total in each figure is given by \( P(n) =\frac {3n^2 - n}{2} \). There is a generalized Pentagonal number which is important in Euler's theory of partitions. It extends the domain to zero and negative numbers.
There are only six numbers that can not be expressed as the sum of four or less pentagonal numbers. They are all less than 100. Can you find them?

Derek Orr points out that 287^2 + 287 + 1 = 82657 is prime.

287 is another number that require four positive squares for their sum. 287 = 15^2 +7^2 + 3^2 + 2^2.

287 is only the fifth number that is an RMS (Root-Mean-Square) Number, the RMS of the divisors is an integer. RMS(287) = sqrt((1^2 + 7^2 + 41^2 + 287^2)/4)=145

Partitions of 40 include ways to select one or more of those numbers that sum to 40. 5+5+10+20 =40, and for that example, it turns out (not by chance) that the sum of their reciprocals, 1/5 + 1/5 + 1/10 + 1/20 = 1, and integer. Now you find one, because according ot OEIS, there are 287 such partitions. Happy Hunting!

The 288th day of the year;
288 is the super-factorial of four. 1! x 2! x 3! x 4! =288. It is important that math students learn not to say this number in public as it is two gross.

288 is also the sum of the first four integers raised to their own power. 288 = 1^1 + 2^2 + 3^3 + 4^4

288 = 2^3 + 4^3 + 6^3

288 is the smallest non-palindrome, non-square, that when multiplied by its reverse is a square: 288 x 882 = 254,016 = 504^2

288^2 + 288 =/- 1 form a pair of twin primes.

288 is the 8th Pentagonal Pyramid number, 1 + 5 + 12 + 22 + 35 + 51 + 70 + 196. ,

288 is a Joy-Giver (Harshad) number, divisible by 18, the sum of its digits.

288 = 2^5 x 3^2, Such numbers with every prime factor to the second, or higher integer is called an Achilles Number.

288 in base 3, 4, 6, and 12 all end in 200.

288 has many factors of two, producing several different expressions as difference of squares, 288 = 73^2 - 71^2 = 38^2 - 34^2 = 22^2 - 14^2 = 17^2 - 1^2

22 can be written as the sum of four squares (including zeros) in 288 ways. 288 = 2^3 + 4^3 + 6^3


The 289th Day of the Year
289 = 15^2 + 8^2

289 is the Hypotenuse of a Primetime Pythagorean Triangle (161, 240 289), which means 289^2 is also the sum of two squares.

There are only two square year days that are neither the sum, or the difference of two primes, 289 is the second. 

The 289th day of the year; 289 is a Friedman number since (8 + 9)2 = 289 (A Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷) and sometimes exponentiation.)Students might try to find the first few multi-digit Friedman numbers.

289 is the square of the prime 17, and the square of the sum of the first four primes, 289 = (2 + 3 + 5 + 7)^2

289 is the largest 3-digit square with increasing digits(amazing, to me, that this occurs so early in the 3-digit numbers.)

289 is the hypotenuse of a primitive Pythagorean triple. Find the legs students!

289 is a 4n+1 prime, which Fermat told us must be the sum of two squares in exactly one way, 8^2 + 15^2 = 289.  

As an odd number, it must also be the difference of two consecutive squares, 145^2 - 144^2 = 289.

Derek Orr points out this pattern about 289:
289 = 17^2.
289 in base 4 = 10201 = 101^2.
289 in base 8 = 441 = 21^2.
289 in base 14 = 169 = 13^2.
289 in base 15 = 144 = 12^2.
289 in base 16 = 121 = 11^2.
289 in base 17 = 100 = 10^2.
289 in base 36 = 81 = 9^2.

289 is one more than twice a square, 2 x 12^2 = 17^2 = 289. Such numbers lead to good approximations of \(\sqrt{2}\). 17/12 = 1.41666... The equation 2x^2 + 1 = y^2 is called a Pell Equation (although only because Euler mis-attributed it.)

The sum of the divisors of 289, 1+17+289 = 307, a prime.It is only the 7th such number, and the last that is a year day.

There are only two year days which are squares and which are neither the sum, nor the difference, of two primes.  289 is the larger, The other is also a three digit number, and a palindrome.

289 is the sum of 4 non-zero 4th powers. 1^4 + 2^4 + 2^4 + 4^4.

The 290th Day of the Year
290 = 13^2 + 11^2 = 17^2 + 1^2.

290 is a sphenic (wedge) number, the product of three distinct primes (290 = 2*5*29).

It is also the sum of four consecutive primes (67 + 71 + 73 + 79) [Students might try to construct and examine a list of numbers that can be written as the sum of two or more consecutive primes]

Derek Orr Pointed out that there are 290 primes below 1900

290 is conjectured to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome.

290 is the tenth prime(29) times ten

290 in base 3 is the concatenation of two three digit palindromes, [101202], and a singl 3-digit palindrome in base twelve, and in base 3, 4, 6, and 12, they all end in the digits202.

290^3 - 290^2 = 4930^2. 290 is only the 17th number for which n^3 - n^2 = y^2. There will only be two more year days that meet that relation and each is one more than a square. And, the sum of the squares of the divisors of 17 is 290.

290^2 + 290 +/- 1 form twin primes. *Derek Orr

290 = 1^4 + 1^4 + 2^4 + 2^4 + 4^4. Can it be done with fewer fourth powers?

The 291st day of the Year
291 is the largest number that is not the sum of distinct non-trivial powers.

ϕ(291)=192 The number of integers less than, and relatively prime to 291 is equal to it's reversal, 192.

291 is also equal to the nth prime + n.... but for which n, children?

The sum of the aliquot divisors of 291 is prime, 101. *Derek Orr

291 appears in five pythagorean triangles, one as the hypotenuse (195, 216, 291) and four more as the shorter leg

291 is a palindrome in base 9(353), and a sequence of three consecutive integers in base 16(123)

291 is a Happy Number, 2^2 + 9^2 + 1^2 = 86, 8^2 + 6^2 = 100, 1^2 + 0^2 + 0^2 = 1. Happy, Happy, Happy!

Because 291 = 6 x 47 + 9, it can be written as the difference of two squares, 291 = 50^2 - 47^2. (An interesting note is that every power of 2^n is either one more or one less than some member of the 6n+9 sequence. The converse is not true, 21 is easy counterexample.

And the 291st digit of pi is a zero.

The 292nd Day of the Year
292 = 74^2 - 72^2, and also 16^2 + 6^2

The continued fraction representation of pi is [3; 7, 15, 1, 292, 1, 1, 1, 2...]; the convergent obtained by truncating before the surprisingly large term 292 yields the excellent rational approximation 355/113=3.4151929 for pi. (113355, divide in the middle, put big over little)
The approximation was found by Chinese mathematician and astronomer Zu Chongzhi(429–500 AD), using Liu Hui's algorithm which is based on the areas of regular polygons approximating a circle*Wik

292 is the number of ways to make change for 1 dollar (or for 1 Euro), using only 1, 5 and 25 cent coins (base five coins).

292! + 291! ± 1 are 595-digit twin primes. (Can you find smaller sums of consecutive factorials like this that are twin primes?)

292 is a palindrome in base 10, base 8(444), and base 7(565), and in base 2(100100100) it repeats the three digits 100, three times (which should explain the 444 in base eight.


The 293rd Day of the Year
293 = 17^2 + 2^2 = 147^2 - 146^2,

293 is the Hypotenuse of a Primitive Triple, (68, 285, 293).

293 is a Sophie Germain Prime. (A prime number p such that 2p + 1 is also prime.) Sophie Germain used them in her investigations of Fermat's Last Theorem. It is an unproven conjecture that there are infinitely many Sophie Germain primes.

293 is also the sum of five cubes, 293=2^3 + + 2^3 + 3^3 + 5^3 + 5^3

and from Jim Wilder @Wilderlab : 293^202 begins with the digits 202 and 202^293 begins with the digits 293.

Prime Curios points out that 300 is kind of prissy, but for math bowlers, the largest prime you can get in the game is 293. Just got to pick off three in the corner on this last roll....

In a normal (non leap year) there are 293 non-prime days in a year.

Another frome Derek Orr, 239 + 2*3*9 = 293.

More from Orr, he is on a roll today: 17 is a prime, and 17^2 + 4 = 293. 293 is a prime, and 293^2 + 4 = 85853, and it is prime. Stop! Go no farther. Only disappointment awaits.

293 is another Happy number, the iteration of the sum or the squares of the digits, arrives at one.

The 294th Day of the Year
292^2 + 293^2 + 294^2 is prime. *Derek Orr

294 is a practical number because all numbers strictly less than 294 can be formed with sums of distinct divisors of 294. There are only 84 such numbers in the year.

294 is the sum of four consecutive squares, 7^2 + 8^2 +9^2 + 10^2

Just learned about "digitally delicate primes," although they are still pretty new in general. In 1978, the mathematician Murray Klamkin wondered if there were any primes so that if you change any digit to any other digit, the newly formed number would be composite. 
 294 is NOT such a number, heck it is not even prime, and there is no year day that is a digitally delicate prime. The smallest one is 294001. If you change any one of these six digits to anything other than what it is you can produce 9^6 = 531,441 different numbers share five of these digits in their present location, and ALL OF THEM ARE COMPOSITE.  Change the 2 in front to a 1, 194001, and of course that is divisible by 3, so try a 3 in front, 394001. That is 47×83×101. Change the first 0 to 2 to get 294101 = 19×23×673 .
The fact is, there seems to be an infinite number of these "digitally delicate darlings."
If that isn't extreme enough for you, somebody decided to extend that to something called "widely digitally delicate" primes. It seems they have somehow proved those are infinitely available too, but I can't tell you the smallest one because, well, nobody has ever found one. (Ok, I'm not sure how they prove there is an infinite number of something that they can't find even one of, but the big brains in math seemed to agree they have. The way they work is that if you imagine an infinite number of zeros in front of a prime, and you change any one of those zeros to something else, "ca-ching", it's composite.
294001 is not one of them, I checked. If you change the zero in front of 0294001 to anything else, it's composite. But if you change the NEXT zero to get 10294001.... bingo, a prime, so the digitally delicate 294001 is NOT WIDELY digitally delicate.
 
Some primes are called unique because no other prime has the same period for its reciprocal (There are only 23 of these known for primes below 10^100. None of these primes contain more than eight different digits. Of all those primes with 8 different digits, has period 294. Notice the similarity to the period of 7 (142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143)*Prime Curios, *Wikipedia

Derek Orr shared that there is no prime in the decade of numbers between 294x10 and 295x10

Found this oddity in my notes: 111152- 2942 = 123,456,789

Quick arithmetic note. Any number evenly divisible by six is the sum of three consecutive integers, 294 = 97+98+99

And I'm writing this in a leap year, 294 + (2x9x4) =366

2(294)+9(294)+4(294) - 1 is 4409, a prime

294 is the only year day (and the only number I know of) which can be written two different ways as the sum of prime numbers using only the digits 1-5 exactly once in the digits of the prime:  294 = 43 + 251 = 53 + 241.  (And the largest year day that can be written in this manner in a single way, is a permutation of the digits of 294, 249.) Students might try to find all the numbers that can be written as the sum of primes using digits 1-6 only once each.

Consider the reciprocal of 294, 1/294 is 42 digits long. SO WHAT? Wikipedia says that there is number with the same length in base ten? Seems incredible.

294 is the area of a Heronian triangle, with all integer sides (can you find it?)

The 295th Day of the Year
295 may be interesting only because it seems to be the least interesting day number of the year. (Willing to be contradicted, send your comments) [Here are several of the best I received from David Brooks: 295 can be partitioned in 6486674127079088 ways. 295 is a 31-gonal number.]
Prime Curios has it as another of those numbers where the sum of the products of the number times each digit is a prime, 2(295) + 9(295) + 5(295)is primes.

And Derek Orr pointed out that "295 is the second proposed Lychrel number." A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proved to exist, but many, including 295, are suspected on heuristic and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend's first name. (Who else thinks he probably mis-spelled her name and when she called him on it, he came up with the idea of a "rough anagram"?  )

Lagrange's Thm says all positive integers can be written with no more than four positive squares. 295 is the 48th of the ones that require the full four, 17^2 + 2^2 + 1^2 + 1^2 works. There are sometimes adjacent numbers that require four (111,112) was the first, but there can never be a triplet.

295 is the sum of four consecutive tetrahedral numbers (the sum of the firs n triangular numbers) 120 + 844 + 56 + 35

295 = 32^2 - 25^2 =148^2 -147^2

The 296th Day of the Year
296 is the number of partitions of 30 with distinct parts. (Even very young students can enjoy exploring the number of partitions of integers, and the difference in the number when the parts must be distinct. The idea can be explored for very young students with number rods, etc)

A cube with an 8x8 checker board on each face has a total of 296 lattice points (where the squares meet)

Derek Orr point out that 296^4 is a ten digit number, but only uses the digits 2,3456 The somewhat famous "look and say" sequence in math, 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, (the second term is 11 because the previous term has One, one; etc) has 296 digits in the 18th term.

Derek Orr points out that 296^4 is a ten-digit number that only uses five consecutive digits, 3, 4, 5, 6, and 7. It ends in 3456. If you add one, everything the same except it ends in 3457, AND, it's a prime. 296 is the 44th year day for which n^4 + 1 is prime.

Both 296 and 297 are of the form p^3 q where p and q are primes. The only other consecutive pair of year days is 135, 136. Because of this they also each have the same number of divisors (8)

Because 296 = 2^3 x 37, the factors of two lead to several expressions of 296 as difference of squares, 75^2 - 73^2 = 296 = 39^2 - 35^2. And as 296 / 4 = 74 = 5^2 + 7^2, 296 = 10^2 + 14^2. (Students might be curious why nothing like that happens with 297.)
296 is also the difference of two cubes, 8^3 - 6^3.

296 is called a refactorable number, it is divisible by the number of divisors it has (8).

The 297th Day of the Year
2972 = 88209 and 88+209 = 297.
Such numbers are called Kaprekar numbers. This is the last day of the year that is a Kaprekar number, named for D R Kaprekar who wrote about them. Every Kaprekar number has a ten-pal (my cutesy variant of a pen-pal) if a n digit number is a Kaprekar number, k, then 10^n - k will also be one. For 297, the ten-pal will be 10^3 - 297 = 703. 703^2= 494209 and 494+209=703.

297 is another 6n+9 number, so it can be written as the difference of two squares, 51^2 - 48^2, and of course, like all odd numbers, it is the difference of two consecutive squares, 149^2 - 148^2.

And this one from * Jim Wilder @wilderlab 2973=26,198,073 and 26+198+073=297. Not a Kaprekar number, but even "Wilder". 297 is only the sixth Kaprekar number. There is only one known number that preserves the relationship in 2nd, 3rd, and fourth power, it is 45.

Both 296 and 297 are of the form p^3 q where p and q are primes. The only other consecutive pair of year days is 135, 136. Because of this they also each have the same number of divisors (8).

Prime Curios points out an interesting relation for 297. If you write a 43 digit number with twenty-nine 7's followed by seven 29's, it's prime. If you reverse that and write seven 29's followed by twenty-nine 7's, still prime.

One from my notes, 29 + 79 + 97 + 92 = 297 (palindrome with only the three digits of n.

The 298th Day of the Year
If you multiply 298 by (298 + 3) you get a palindromic number, 89,698. Can every number be similarly adjusted to make a palindrome? And this one is not just a Palindrome, it's a strobogrammatic one, rotate it 180 degrees and you get another palindrome, 86968.  (Some restrict the term stobogram only to numbers that recreate themselves after rotation, and prefer ambigram for the ones that rotate to make a different number.)

298 = \( {12 \choose 1} + {12 \choose 2} + {12 \choose 3} \) This is related to the Egg Drop numbers

6 x 298 +/- 1 are twin primes. The 55th number of the year for which this is true.

298 and 299 have the same number of divisors. All semi-primes(product of two primes) have four divisors. 298 is the 45th number that has the same number of divisors as it's successor.


The 299th Day of the Year
If a cubic cake was cut with 12 straight cuts, it can produce a maximum of 299 pieces.... a good day to "let 'em eat cake."

An annulus (2-D donut) would require 23 cuts to produce the same number . There are 299 semiprimes less than 1000 which are products of two primes.

and 299 is a semi-prime, but with both factors having the same length (number of digits). Only four year days left share this property.

299 = 150^2 - 149^2.

299 = 99 + 1 + 99 + 1 + 99

299 is the long leg of a Pythagorean triangle. (180, 299, 349)


The 300th Day of the Year
300 is a triangular number, the sum of the integers from 1 to 24.

300 is also the sum of a pair of twin primes (149 + 151). And the sum of ten consecutive primes, 300 = 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.

300 = 2 x 5 x 3 x 5 x 2, palindrome product of primes.

300 is the largest natural number that is Not the sum of a prime and a 3-almost primes (product of three primes, not necessarily distinct). 7+2 x 3 x 5 = 37 can be so expressed. Students might find other numbers that are no expressible as such.*Prime Curios

300 is is a palindrome in three consecutive bases, base 7(606), base 8 (454), and base 9 (363). And in Roman Numerals it is a repdigit, CCC

300 can be expressed as the difference of two squares in three different ways, 76^2 - 74^2 = 28^2 - 22^2 = 20^2 - 10^2 = 300

Oh Dear! 300 is not a Happy Number. Iterations of the sum of the squares of the digits fall into the eight- cycyle 3 --- 9 --- 81 --- 65 --- 61 --- (37 --- 58 --- 89 --- 145 --- 51 ---26 --- 40 --- 16 ----37)

300 is the 49th Day of the Year for which n^2 + 1 is prime.

The Fibonacci sequence Modula 50, has a period length of 300. As an example for a smaller number, mod 2, the numbers 1, 1, 2, 3, 5, 8, 13 Mod 2 have residues 1, 1, 0, 1, 1, 0, 1, for a repeating pattern of 3.

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