## Thursday, May 31, 2012

### 185

185 = 5 x 37.

185 has two representations as a sum of two squares: 185 = 42 + 132 = 82 + 112.

185 is the hypotenuse of two primitive Pythagorean triples: 1852 = 572 + 1762 = 1042 + 1532.

185 is the number of conjugacy classes in the automorphism group of the 8-dimensional hypercube.

185 is 505 in base 6 and 353 in base 7.

NGC 185 is a dwarf elliptical galaxy in the constellation Cassiopeia.

## Wednesday, May 30, 2012

### 274

274 = 2 x 137.

274 has a representation as a sum of two squares: 274 = 72 + 152.

274 is a centered triangular number.

274 is the maximum number of regions into which 17 circles divide the plane.

(274 - 2)2 + (274 - 7)7 + (274 - 4)4 is a prime.

The 274th Lucas number plus 274 is a prime.

274 is the sum of five positive cubes: 274 = 3(23) + 2(53).

Source: Prime Curios!

## Tuesday, May 29, 2012

### 1375

1375 = 5 x 5 x 5 x 11.

1375 is a decagonal pyramidal number (A007585).

1375 is 111133 in base 4.

1375 is the smaller of two consecutive numbers each divisible by a cube (A068140). It is also the smallest of three consecutive numbers divisible by a square (A070258).

13752 = 1,890,625 contains exactly seven different digits (A054035).

## Friday, May 25, 2012

### 735

735 = 3 x 5 x 7 x 7. It is a number divisible by the product of its digits (A007602). It is also divisible by the sum of its digits (7 + 3 + 5 = 15) (A038186).

735 is the smallest number that is the concatenation of its distinct prime factors.

735 is 3223 in base 6 (A029953).

735 is the sum of nine positive fifth powers (A003354): 735 = 35 + 35 + 35 + 15 + 15 + 15 + 15 + 15 + 15.

Source: Number Gossip

### 723

723 = 3 x 241.

723 = (1!)! + (2!)! + (3!)! (A059590).

723 = 36 - 6.

723 is 333 in base 15.

723 is the smallest positive number that can be written as a sum of distinct Fibonacci numbers in 25 ways (A013583).

723 is the sum of fifth powers of primes (A122616): 723 = 35 + 15(25).

## Thursday, May 24, 2012

### 1134

1134 = 2 x 3 x 3 x 3 x 3 x 7.

1134 is the smallest number such that the sum of its proper divisors equals the sum of the sums of proper divisors of its proper divisors.

1134 is the number of permutations of nine items that fix five elements.

1134 is 14014 in base 5.

1134 is divisible by the number of primes below it (189): 1134/189 = 6.

1134 is a concentric 14-gonal (tetradecagonal) number (A195145).

Source: Number Gossip

## Wednesday, May 23, 2012

### 916

916 = 2 x 2 x 229.

When 916 is raised to the sum of its digits and increased by 1, the result is a prime: 916(9 + 1 + 6) + 1. It is the largest such three-digit number.

916 has a representation as a sum of two squares: 916 = 42 + 302.

916 is a strobogrammatic number (A000787). It appears the same whether it is viewed normally or upside down.

The sum of the digits of 916 (9 + 1 + 6 = 16) is a substring of 916 (A052018).

916 America is an asteroid discovered in 1915 by G.N. Neujmin. There are seven letters in the asteroid's name and seven letters in its discoverer's last name. Note that 916 * 1915 + 7 * 7 is a prime.

Source: Prime Curios!

## Tuesday, May 22, 2012

### 715

715 = 5 x 11 x 13.

715 is the tenth pentatope number. It is also a pentagonal number (A000326), a hendecagonal number (A051682), and a hexagonal pyramidal number (A002412).

715 is 222111 in base 3, 23023 in base 4, and 1313 in base 8.

715 is the number of intersections of diagonals in the interior of a regular 13-gon (A006561).

715 is the smallest positive number that can be written as a sum of distinct Fibonacci numbers in 26 ways.

Source: Number Gossip

## Monday, May 21, 2012

### 134

134 = 2 x 67.

1342 - 672 = 13467.

134 is the maximum number of regions into which 12 circles divide the plane.

134 is 11222 in base 3 and 112 in base 11.

134 is the sum of three distinct positive cubes (A024975): 134 = 53 + 23 + 13. It is also the sum of five positive cubes (A003328): 134 = 43 + 33 + 33 + 23 + 23.

Source: Prime Curios!

## Friday, May 18, 2012

### 308

308 = 2 x 2 x 7 x 11.

308 is a heptagonal pyramidal number.

308 is the maximum number of regions into which 18 circles divide the plane.

3083 + 3080 + 3088 is a prime number.

308 is the sum of two consecutive primes: 308 = 151 + 157.

308 is 102102 in base 3. It is 464 in base 8.

Source: Prime Curios!

## Thursday, May 17, 2012

### 704

704 = 2 x 2 x 2 x 2 x 2 x 2 x 11.

704 has a unique representation as a sum of three squares: 704 = 82 + 82 + 242.

704 is the maximum number of regions into which 27 circles divide the plane. It is also a central polygonal number (the maximum number of pieces formed when slicing a pancake with 37 cuts) (A000124).

704 is the number of one-sided octominoes (A000988).

The sum of the fourth power of the prime factors of 704 is a prime (A134620): 6(24) + 114 = 14737.

## Wednesday, May 16, 2012

### 496

496 = 2 x 2 x 2 x 2 x 31.

496 is the 31st triangular number. It is the sum of all integers from 1 to 31. It is also a hexagonal number and a centered nonagonal number.

496 is the smallest triangular number such that the sum of its digits cubed is prime: 43 + 93 + 63 = 1009.

496 is the third perfect number, associated with the Mersenne prime 31. It is the largest perfect number with all different digits. It is the smallest perfect number such that the sum of its digits is prime: 4 + 9 + 6 = 19.

496 = 13 + 33 + 53 + 73.

Source: Number Gossip

## Tuesday, May 15, 2012

### 2219

2219 = 7 x 317. It is the product of two primes whose sum is a perfect square (A141755): 7 + 317 = 324 = 182.

2219 is the number of 14-hexes with reflectional symmetry (A002215). It is also the number of restricted hexagonal polyominoes with 7 cells (A002212).

2219 is the number of incongruent ways to tile a 5 x 66 room with 1 x 2 Tatami mats, with at most three Tatami mats meeting at a point (A068930).

2219 is a number of the form ab + cd, where a, b, c, and d are the first four primes (A168349): 2219 = 25 + 37.

2217 = 3 x 739, 2218 = 2 x 1109, and 2219 are all semiprimes (A115393).

## Monday, May 14, 2012

### 280

280 = 2 x 2 x 2 x 5 x 7.

280 is 100011000 in base 2 (binary).

280 has a unique representation as a sum of three squares: 280 = 62 + 102 + 122.

The sum of the first 280 consecutive primes mod 280 is prime.

280 is the number of ways 18 people around a round table can shake hands in a non-crossing way, up to rotation.

280 is an octagonal number.

Source: Prime Curios!

## Friday, May 11, 2012

### 578

578 = 2 x 17 x 17.

578 has two representations as a sum of two squares (A118882): 578 = 72 + 232 = 172 + 172.

578 is the number of graphs with 7 vertices with clique number 3.

578 is 210102 in base 3 and 21002 in base 4. It is 242 in base 16 and 123 in base 23.

578 is the smallest of four consecutive integers divisible by four consecutive primes respectively (A072555).

## Thursday, May 10, 2012

### 970

970 = 2 x 5 x 97.

970 has two representations as a sum of two squares: 970 = 32 + 312 = 212 + 232.

970 is a heptagonal number (A000566).

970 is the number of connected graphs with 8 vertices and 17 edges.

970 is 33022 in base 4 and 12340 in base 5.

The sum of the divisors of 970 is a square (A006532): 1 + 2 + 5 + 10 + 97 + 194 + 485 + 970 = 1764 = 422.

## Wednesday, May 9, 2012

### 390

390 = 2 x 3 x 5 x 13. The prime factorization of 390 exhibits the smallest digital sum, [(2 + 3 + 5 + (1 + 3)] = 14, of any factorization with four primes with distinct digital sums (2, 3, 5, and 4).

390 is the number of integer partitions of 32 into distinct parts.

390 is the sum of two consecutive primes: 390 = 193 + 197. It is also the sum of four consecutive primes: 390 = 89 + 97 + 101 + 103.

390 is the 12th term in the sequence involving rooted polygonal cacti (Husimi graphs) with n nodes.

390 is 12012 in base 4 and 3030 in base 5. It is 606 in base 8.

Source: Prime Curios!

## Tuesday, May 8, 2012

### 664

664 = 2 x 2 x 2 x 83.

664 is a value of n such that n(n + 7) is a palindrome (445544).

664 is 554 in base 11 and 474 in base 2.

664 is a generalized heptagonal number (A085787).

664 is the solution to the postage stamp problem for 6 denominations and 7 stamps (A001211).

664 is the sum of three distinct positive cubes (A024975): 664 = 83 + 53 + 33.

## Monday, May 7, 2012

### 471

471 = 3 x 157.

471 is 111010111 in base 2 (binary) (A006995). It is 333 in base 12 and 727 in base 8.

471 is the smallest number with the property that its first four multiples (471, 942, 1413, 1884) contain the digit 4.

471 is the sum of three consecutive primes (A034961): 471 = 151 + 157 + 163.

471 is the sum of twelve positive fifth powers (A003357): 471 = 35 + 25 + 25 + 25 + 25 + 25 + 25 + 25 + 15 + 15 + 15 + 15.

## Friday, May 4, 2012

### 955

955 = 5 x 191.

955 is the number of ways to arrange the numbers 1 to 9 around a circle so that the sums of adjacent numbers are distinct.

955 is 32323 in base 4 and 353 in base 17.

955 and 9552 = 912025 have the same digit sum (A058369): 9 + 5 + 5 = 19 = 9 + 1 + 2 + 0 + 2 + 5. They also have the same initial digit and final digit (A086457).

955 and the 955th prime (7537) have only the digit 5 in common (A107936).

## Thursday, May 3, 2012

### 465

465 = 3 x 5 x 31.

465 is the 30th triangular number (the sum of all integers from 1 to 30).

465 is a Kaprekar constant in base 2.

465 x 831 = 386415 (the product has the same digits as its factors).

465 is 393 in base 11.

465 is a member of the Padovan sequence.

Source: Zoo of Numbers

## Wednesday, May 2, 2012

### 943

943 = 23 x 41.

943*349! - 1 is a titanic prime.

943 is a Lucas 6-step number.

943 and the 943rd prime (7451) have only the digit 4 in common (A107935).

943 is the sum of 18 and no fewer nonzero fourth powers (A046049): 943 = 44 + 34 + 34 + 34 + 34 + 34 + 34 + 34 + 34 + 24 + 24 + 14 + 14 + 14 + 14 + 14 + 14 + 14.

Source: Prime Curios!

## Tuesday, May 1, 2012

### 365

365 = 5 x 73.

365 is 101101101 in base 2 (binary). It is 555 in base 8.

365 is a centered square number and a centered tridecagonal number (A051865).

365 is the smallest number that can be written as a sum of consecutive squares in more than one way: 365 = 142 + 132 = 122 + 112 + 102.

365 has two representations as a sum of two squares: 365 = 22 + 192 = 132 + 142.

365 is the hypotenuse of two primitive Pythagorean triples: 3652 = 272 + 3642 = 762 + 3572.

365 is the most frequent number of days in a year.