## Friday, August 31, 2012

### 788

788 = 2 x 2 x 197.

788 has a representation as a sum of two squares: 788 = 22 + 282.

788 is the smallest of six consecutive numbers divisible by six consecutive primes: 788 is divisible by 2, 789 by 3, 790 by 5, 791 by 7, 792 by 11, 793 by 13.

788 is 404 in base 14.

7882 is the sum of two successive primes (A074924): 7882 = 620944 = 310463 + 310481.

Source: Prime Curios!

## Thursday, August 30, 2012

### 236

236 = 2 x 2 x 59.

The product of the digits (2 x 3 x 6 = 36) of 236 is the reverse of the sum of its prime factors (2 + 2 + 59 = 63).

236 is the number of possible moves in Othello (Reversi) after two moves by both players.

236 is the sum of 12 consecutive primes: 236 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41.

236 is 282 in base 9.

Source: Prime Curios!

## Wednesday, August 29, 2012

### 155

155 = 5 x 31.

155 has the representation 155 = 27 + 27.

155 = 22 + 3! + 5! + 72 -11 - 13.

155 is the only three-digit number such that 155155 - 155 + 1 is prime.

155 is the sum of its prime factors plus the primes between these two prime factors: 155 = 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31.

155 = 33 + 43 + 43.

155 = 51 + 52 + 53.

Source: Prime Curios!

## Tuesday, August 28, 2012

### 872

872 = 2 x 2 x 2 x 109.

872 has a representation as a sum of two squares: 872 = 142 + 262.

872 is the maximum number of regions into which 30 circles can divide the plane.

872! + 1 is a factorial prime.

872 is 464 in base 14.

Source: Prime Curios!

## Monday, August 27, 2012

### 682

682 = 2 x 11 x 31.

682 is 1010101010 in base 2 (binary). It is 22222 in base 4 (A014192) and 262 in base 17.

682 x 682! + 1 is a prime number.

682 is the sum of four consecutive primes (A034963): 682 = 163 + 167 + 173 + 179. It is also the sum of 10 consecutive primes (A127337): 682 = 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89.

682 is the sum of 12 positive fifth powers (A003357): 682 = 35 + 35 + 25 + 25 + 25 + 25 + 25 + 25 + 15 + 15 + 1+ 15.

Source: Prime Curios

## Friday, August 24, 2012

### 584

584 = 2 x 2 x 2 x 73.

584 is the number of ways to color the vertices of a triangle with 12 colors, up to rotation.

584 has a representation as a sum of two squares: 584 = 102 + 222.

584 is 248 in base 16.

584 is the sum of three distinct positive cubes (A024975): 584 = 83 + 43 + 23.

## Thursday, August 23, 2012

### 1612

1612 = 2 x 2 x 13 x 31.

1612 has a unique representation as a sum of three squares: 1612 = 62 + 262 + 302.

1612 is a divisor of 256 - 1.

1612 is 22422 in base 5 and 11244 in base 6. It is 727 in base 15.

1612 is the number of primes between 214 and 215 (A036378).

1612 is the sum of the first 33 semiprimes (A062198).

Source: On-Line Encyclopedia of Integer Sequences

## Wednesday, August 22, 2012

### 1521

1521 = 3 x 3 x 13 x 13.

1521 is a perfect square: 1521 = 392.

1521 is the smallest number that can be written as the sum of four distinct cubes in three ways: 1521 = 113 + 53 + 43 + 13 = 103 + 83 + 23 + 13 = 93 + 83 + 63 + 43.

1521 has a representation as a sum of two squares: 1521 = 152 + 362.

1521 is a divisor of 223 - 1.

1521 is a centered octagonal number.

## Tuesday, August 21, 2012

### 1413

1413 = 3 x 3 x 157.

1413 is the smallest number that cannot be formed using the digits 0 to 7 at most once, together with the arithmetic operations (symbols) +. -. *, and / (A060315).

1413 has a representation as a sum of two squares: 1413 = 182 + 332.

1413 is 999 in base 12 (A048336). It is 585 in base 16.

1413 has the representation 1413 = 2 x 36 - 45.

## Monday, August 20, 2012

### 295

295 = 5 x 59.

2 x 295 + 9 x 295 + 5 * 295 + 1 is a prime number (4271).

295 is a structured deltoidal hexecontahedral number.

295 is a centered tetrahedral number (A005894).

295 is 171 in base 14.

295 is the number of primes less than 442 = 1936 (A038107).

The sum of the digits of 295 is a square (A028839): 2 + 9 + 5 = 16 = 42.

Source: Prime Curios!

## Friday, August 17, 2012

### 385

385 = 5 x 7 x 11.

385 is the number of integer partitions of 18.

385 is a square pyramidal number (the sum of the squares of all integers from 1 to 10).

385 is a Cullen number.

385 = 1932 - 1922.

385 is 1441 in base 6 and 181 in base 18.

## Thursday, August 16, 2012

### 193

193 is a prime number.

191 and 193 form a twin prime pair.

193 can be formed from the sum of the products of the first three twin prime pairs: 193 = 3 * 5 + 5 * 7 + 11 * 13.

193 can be written as the difference between the product and the sum of the first four primes: 193 = 2 * 3 * 5 * 7 - (2 + 3 + 5 + 7).

193 is the smallest prime whose reversal is a Smith number.

193 is the smallest prime whose fifth power (267785184193) contains all the digits from 1 to 9.

193 has a representation as a sum of two squares: 193 = 72 + 122.

193 is the hypotenuse of a primitive Pythagorean triple: 1932 = 952 + 1682.

Source: Prime Curios!

## Wednesday, August 15, 2012

### 569

569 is a prime number.

569 and 571 form a twin prime pair. The 569th and 571st Fibonacci numbers are also primes.

569 = 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92.

569 is 2345 in base 6.

569 has a representation as a sum of two squares: 569 = 132 + 202.

569 is the hypotenuse of a primitive Pythagorean triple: 5692 = 2312 + 5202.

Source: Prime Curios!

## Tuesday, August 14, 2012

### 449

449 is a prime number.

449 is the smallest prime number whose digits are composite perfect squares. It is also the smallest prime number that can be constructed from two concatenated prime squares: 22 and 72. And it is the smallest prime whose digits are all semiprimes.

449 is the 87th prime. Note that 42 + 42 + 92 = 82 + 72 = 113.

449 is the sum of five consecutive prime numbers: 449 = 79 + 83 + 89 + 97 + 101.

449 has a representation as a sum of two squares: 449 = 72 + 202.

449 is the hypotenuse of a primitive Pythagorean triple: 4492 = 2802 + 3512.

Source: Prime Curios!

## Monday, August 13, 2012

### 932

932 = 2 x 2 x 233.

932 has a representation as a sum to two squares: 932 = 162 + 262.

932 is the maximum number of regions into which 31 circles divide the plane.

932 is the number of permutations of length 9 within distance 2 of a fixed permutation (A002524).

932 is the sum of five distinct positive pentatope numbers (A104396).

Source: On-Line Encyclopedia of Integer Sequences

## Friday, August 10, 2012

### 1305

1305 = 3 x 3 x 5 x 29.

1305 has two representations as a sum of two squares: 1305 = 32 + 362 = 242 + 272.

1305 is the number of graphs with 11 vertices and 9 edges.

1305 is 909 in base 12.

1305 is a number n such that n2 (1703025) can be obtained by inserting internal (but not necessarily contiguous) digits (A046851).

## Thursday, August 9, 2012

### 6115

6115 = 5 x 1223.

6115 is a number n such that n2 - 6 and n2 + 6 are both prime (A108403).

6115 is the number of S, E, and NW-moving king's tours on a 13 x 13 board summed over all starting positions (A187510).

6115 Martinduncan is an asteroid discovered in 1984 and named in honor of astronomer Martin J. Duncan of Queen's University, Kingston, Ontario.

Source: On-Line Encyclopedia of Integer Sequences

## Wednesday, August 8, 2012

### 624

624 = 2 x 2 x 2 x 2 x 3 x 13.

624 is 4444 in base 5. It is 1551 in base 7.

624 is the smallest number with the property that its first five multiples (624, 1248, 1872, 2496, 3120) contain the digit 2.

624 is the sum of twin primes: 624 = 311 + 313.

624 Hektor is the largest of the Jupiter Trojan asteroids.

## Tuesday, August 7, 2012

### 729

729 = 3 x 3 x 3 x 3 x 3 x 3 = 36. It is a perfect sixth power.

729 is a centered octagonal number.

and 729 are consecutive Smith numbers, so they are known as Smith brothers.

729 is the smallest odd cube represented as the sum of two prime numbers: 729 = 2 + 727.

729 is a cube that is the sum of three cubes: 729 = 93 = 13 + 63 + 83.

729 = 272 = (1 + 2 + 2 + 2 + 4 + 4 + 4 + 8)2 = 13 + 23 + 23 + 23 + 43 + 43 + 43 + 83.

729 is 1331 in base 8 and 121 in base 26.

Source: Prime Curios!

## Monday, August 6, 2012

### 618

618 = 2 x 3 x 103.

618 = 0!2 + 1!2 + 2!2 + 3!2 + 4!2.

618 is 4433 in base 5.

618 is the sum of 10 positive fifth powers (A003355): 618 = 35 + 35 + 25 + 25 + 25 + 25 + 15 + 15 + 15 + 15.

618 is the number of ternary squarefree words of length 15 (A006156).

## Friday, August 3, 2012

### 566

566 = 2 x 283.

566 is the number of ways to place 24 points on a 12 x 12 grid so that no three points are on a line (A000769).

The sum of the fourth power of the prime factors of 566 is a prime (A134620): 24 + 2834 = 6414247937, a prime.

566 is the sum of five distinct pentatope numbers (A104395).

566 is the sum of the first 20 primes minus the next prime (A166448).

## Thursday, August 2, 2012

### 409

409 is a prime number.

409 is a centered triangular number.

409 has a representation as a sum of two squares: 409 = 32 + 202.

409 is the hypotenuse of a primitive Pythagorean triple: 4092 = 1202 + 3912.

409 is the smallest prime that, along with both its prime neighbors, uses non-prime digits: 401, 409, 419.

409 is the number of primes less than 10,000 that form members of twin prime pairs.

Source: Prime Curios!

## Wednesday, August 1, 2012

### 815

815 = 5 x 163.

815 is a Lucas 3-step number.

815 is 3435 in base 6.

8152 is the sum of three consecutive primes (A076304).

815 and 817 are both semiprimes (A136196).