## Friday, June 29, 2012

### 722

722 = 2 x 19 x 19.

722 has a representation as a sum of two squares: 722 = 192 + 192.

722 is the sum of the fourth powers of the first three primes (A133531): 722 = 24 + 34 + 54.

722 is the sum of two distinct factorials, the sum of the factorials of the digits of 26 (A061602): 722 = 2! + 6!

722 is the maximum number of regions into which the plane can be divided by 10 (concave) quadrilaterals (A077591).

## Thursday, June 28, 2012

### 151

151 is a prime number.

149 and 151 form a twin prime pair.

151 is a centered decagonal number.

151 is the smallest palindromic prime sandwiched between two consecutive squareful (or non-squarefree) numbers.

151 is the smallest prime that begins a 3-run of sums of five consecutive primes: 151 + 157 + 163 + 167 + 173 = 811; and 811 + 821 + 823 + 827 + 829 = 4111, and 4111 + 4127 + 4129 + 4133 + 4139 = 20639. Can you find the smallest 4-run example?

The Statue of Liberty, from its base to the tip of its torch, was designed to be 151 feet tall.

Source: Prime Curios!

## Wednesday, June 27, 2012

### 231

231 = 3 x 7 x 11. It is the 22nd number that is the product of three distinct primes.

231 is 11100111 in base 2 (binary).

231 is the number of integer partitions of 16.

231 is the 21st triangular number. It is the smallest triangular number in which the sum of the external digits equals the sum of the internal digits.

231 is a hexagonal number and an octahedral number.

231 = 22 + 32 + 72 + 132.

The U.S. liquid gallon is defined as 231 cubic inches.

Source: Number Gossip

## Tuesday, June 26, 2012

### 913

913 = 11 x 83.

The integer digits of 9132 = 833569 are exactly the same (albeit in a different order) as the integer digits of 9142 = 835396 (A072841).

The digit sum of 913 equals the digit sum of its prime factors (A036920).

913 is a generalized polygonal number (A080859).

913 is the sum of the integers from 1 to 43, excluding perfect fifth powers (A132336).

Source: On-Line Encyclopedia of Integer Sequences

## Monday, June 25, 2012

### 318

318 = 2 x 3 x 53.

The sum of the digits of 318 is one less than the sum of the digits of its prime factors.

318 is the number of unlabeled partially ordered sets of 6 elements.

318 is the sum of 12 consecutive primes: 318 = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.

318 is 2233 in base 5. It is 383 in base 9.

Source: Prime Curios!

## Friday, June 22, 2012

### 523

523 is a prime number.

521 and 523 form a twin prime pair.

523 together with 541 form the smallest two consecutive primes such that the sums of their digits are equal. It is the smallest prime that is followed immediately by 17 composite numbers.

523 is the only three-digit prime containing all three of the first three prime digits. It is the largest prime with a prime number of distinct prime digits. It is the only prime formed from three consecutive primes, one being the sum of the other two.

523 is 313 in base 13.

523 is the sum of seven consecutive primes: 523 = 61 + 67 + 71 + 73 + 79 + 83 + 89.

Source: Number Gossip

## Thursday, June 21, 2012

### 1651

1651 = 13 x 127.

1651 is 11001110011 in base 2 (binary) (A048700).

1651 is a heptagonal number (A000566).

1651 is a potential magic constant of a 7 x 7 magic squares composed of consecutive primes (A188536).

1651 is the sum of eight but not fewer squared primes (A183215).

Source: On-Line Encyclopedia of Integer Sequences

## Wednesday, June 20, 2012

### 7698

7698 = 2 x 3 x 1283.

76982 = 59259204.

7698 is the number of ways to place 10 non-attacking kings on a 4 x 10 chessboard (A061593).

The set of base 7 digits of 7698 equals the set of base 9 digits of 7698 (A037439): {0, 1, 3, 5}.

7698 is a number n for which 12n + 1, 12n + 5, 12n + 7, and 12n + 11 are primes (A123985).

Source: On-Line Encyclopedia of Integer Sequences

## Tuesday, June 19, 2012

### 765

765 = 3 x 3 x 5 x 17.

765 has two representations as a sum of two squares: 765 = 62 + 272 = 182 + 212.

765 = 401 + 263 + 89 + 7 + 5. Its reversal, 567, is the smallest number that is the sum of different primes using all ten digits.

765 is 1011111101 in base 2 (A006995) and 636 in base 11.

765 is the sum of eight positive fifth powers (A003353): 765 = 35 + 35 + 35 + 25 + 15 + 15 + 15 + 15.

Source: Prime Curios!

## Monday, June 18, 2012

### 204

204 = 2 x 2 x 3 x 17.

204 is 11001100 in base 2 (binary).

204 is a nonagonal number and a square pyramidal number. It is also an abundant number.

The number 204n + 1 is prime for n = 2, 0, and 4.

204 is the square root of the triangular number 41616.

2042 = 233 + 243 + 253 = 41616.

204 is the sum of twin primes: 204 = 101 + 103.

204 is the number of different squares that you can find on a standard chessboard.

Source: Number Gossip

## Friday, June 15, 2012

### 1518

1518 = 2 x 3 x 11 x 23.

1518 is a divisor of 4710 - 1.

1518 is the sum of its proper divisors that contain the digit 5: 1518 = 253 + 506 + 759.

1518 is the sum of two consecutive sums of consecutive numbers: 1518 = 121 + 122 + . . . + 131 + 132 = 133 + 134 + . . . + 142 + 143.

1518 is 2002020 in base 3. It is 330 in base 22.

## Thursday, June 14, 2012

### 758

758 = 2 x 379.

758 is the maximum number of regions into which 28 circles divide the plane.

758 is 464 in base 13 and 262 in base 18.

758 is the number of walks on a cubic 4 x 4 x 4 lattice (starting at the origin and not going below the x-y plane) (A005573).

758 is the smallest of three consecutive integers (758, 759, 760) divisible by three consecutive primes respectively (2, 3, 5) (A073607).

Source: On-Line Encyclopedia of Integer Sequences

## Wednesday, June 13, 2012

### 391

391 = 17 x 23.

391 is a Smith number (A0906753). The sum of its digits (3 + 9 + 1 = 13) equals the sum of the digits of its prime factors (1 + 7 + 2 + 3 = 13).

391 is a centered pentagonal number (A005891).

391 is 474 in base 9 (A029955).

391 is a substring of 2391 (A032740).

391, 393, and 395 are all semiprimes (A092125), between the prime numbers 389 and 397.

Source: On-Line Encyclopedia of Integer Sequences

## Tuesday, June 12, 2012

### 473

473 = 11 x 43.

473 is the largest known number whose square (223729) and fourth power (50054665441) use different digits (and no digit 8).

473 x 8 = 3784 (same digits are used on both sides).

473 is 575 in base 9.

473 is the sum of five consecutive primes (A034964): 473 = 83 + 89 + 97 + 101 + 103.

473 is the number of ways of making change for 51 cents using coins of 1, 2, 5, 10, 20, and 50 cents (A001313).

## Monday, June 11, 2012

### 965

965 = 5 x 193.

965 has two representations as a sum of two squares: 965 = 22 + 312 = 172 + 262
(A088909).

965 is the hypotenuse of two primitive Pythagorean triples (A024409): 9652 = 1242 + 9572 = 3872 + 8842 (A165158).

Both 965 and 9652 = 931225 have the same initial digit and the same final digit (A086457).

965 is the number of unit squares at least 50% covered by a circle inscribed in an integer square of size 35 x 35 (A051233).

Source: On-Line Encyclopedia of Integer Sequences

## Thursday, June 7, 2012

### 1400

1400 = 2 x 2 x 2 x 5 x 5 x 7.

1400 is a divisor of 992 - 1.

1400 is the number of different arrangements of 4 non-attacking queens on a 4 x 10 chessboard (A061990).

1400 is 4040 in base 7.

1400 is divisible by the square of the sum of its digits (25) (A072081).

1400 is the sum of the digits in the 41st row of Pascal's triangle (A066600).

## Wednesday, June 6, 2012

### 656

656 = 2 x 2 x 2 x 2 x 41.

656 is 1010010000 in base 2 (binary). It is a palindrome in base 3 (220022) (A007633 and A014190), base 9 (808), and base 10 (656) (A029965).

656 has a representation as a sum of two squares: 656 = 162 + 202.

656 has 10 binary digits and 10 divisors (1, 2, 4, 8, 16, 41, 82, 164, 328, 656) (A135772).

656 plus the 656th prime produces a triangular number (A115882): 656 + 4909 = 5565.

## Tuesday, June 5, 2012

### 848

848 = 2 x 2 x 2 x 2 x 53.

848 has a representation as a sum of two squares: 848 = 82 + 282.

848 is the number of inequivalent binary linear codes of length 9.

848 is the sum of the first 33 nonprimes (A051349).

848 is the sum of distinct factorials (A059590): 848 = 6! + 5! + 3! + 2!

## Monday, June 4, 2012

### 938

938 = 2 x 7 x 67.

938 is the number of lines passing through at least two points of an 8 x 8 grid of points (A018808).

938 is the total number of positive integers below 105 requiring two positive cubes in their representation as a sum of cubes (A181375).

938 is the smallest of three consecutive integers (938, 939, 940) divisible by three consecutive primes respectively (2, 3, 5) (A073607).

938 is the sum of two consecutive primes and all the composite numbers in between (A174518): 938 = 131 + 132 + 133 + 134 + 135 + 136 + 137.

## Friday, June 1, 2012

### 191

191 is a prime number. It is a Sophie Germain prime because 2 x 191 + 1 = 383 is also a prime. In addition, the resulting number is a palindrome.

191 and 193 form a twin prime pair.

191 is the maximum number of regions into which 19 lines divide the plane.

191 is the smallest palindromic prime whose digit sum (1 + 9 + 1 = 11) is a multidigit palindromic prime. It is also the smallest multidigit palindromic prime that yields a palindrome when multiplied by the next prime: 191 x 193 = 36863.

191 is the smallest prime in a set of four consecutive primes (191, 193, 197, 199) whose sum of digits is another set of four primes (11, 13, 17, 19).

The common U.S. coins--a silver dollar, half-dollar, quarter, dime, nickel, and penny--add up to 191 cents.

Source: Number Gossip