## Monday, December 24, 2012

### 2013

2013 = 3 x 11 x 61.

2013 is a permutation of the digits 0, 1, 2, and 3  (A203569)

2013, 2014, and 2015 have the same number of divisors (A005238). Each is the product of three distinct primes (A066509 and A113789).

The digital sums of 2013 in base 2 (11111011101), base 3 (2202120), and base 5 (31023) are equal (A135121).

The sum of 2013 and its prime factors equals the sum of 2014 and its prime factors (A020700): 2013 + 3 + 11 + 61 = 2014 + 2 + 19 + 53 = 2088.

2013 is the number of trees with 27 nodes and 4 leaves (A055291).

The year 2013 celebrates the Mathematics of Planet Earth.

2013 is also the International Year of Statistics.

## Friday, December 21, 2012

### 931

931 = 7 x 7 x 19.

931 is the sum of three consecutive prime numbers: 931 = 307 + 311 + 313. It is also the sum of 11 consecutive primes (A127338).

931 is 777 in base 11 and 111 in base 30. It is 2500 in base 7.

931 is a divisor of 303 - 1.

931 is a central polygonal number (A002061).

931 is a number that is the sum of exactly three sets of Fibonacci numbers (A122195).

Source: On-Line Encyclopedia of Integer Sequences

## Thursday, December 20, 2012

### 719

719 is a prime number. It is a Sophie Germain prime because 2 x 719 + 1 = 1439, also a prime.

719 is a number that can be expressed as both a factorial prime (6! - 1) and as an abundant number less 1.

719 is the number of rooted trees with 12 vertices.

719 is a right-truncatable prime number: 719, 71, and 7 are all primes.

719, 197, and 971 are all primes.

The hour and second hands of an analog clock align 719 times every twelve hours.

Source: Prime Curios!

## Wednesday, December 19, 2012

### 1234

1234 = 2 x 617.

1234 is the smallest four-digit number with increasing digits (A007908).

1234 is the number of digits in the 12th Fermat number (A057755).

1234 is 103102 in base 4. It is 14414 in base 5 and 5414 in base 6.

1234 is the number of ways to place three nonattacking bishops on a 3 x 8 board (A172207).

1234 has a representation as a sum of two squares: 1234 = 32 + 352.

## Tuesday, December 18, 2012

### 541

541 is a prime number. It is the 100th prime.

541 is a centered dodecagonal number. It is the 10th hexagonal star number.

5! + 4! + 1 = 145, the reverse of 541. 541 is the only odd number greater than 1 with this property.

541 is the number of orderings of five objects with ties allowed.

541 has a representation as a sum of two squares: 541 = 102 + 212.

541 is the hypotenuse of a primitive Pythagorean triple: 5412 = 3412 + 4202.

541 has the representation 541 = 29 + 29.

Source: Number Gossip

## Monday, December 17, 2012

### 1000

1000 = 2 x 2 x 2 x 5 x 5 x 5. 1000 is a perfect cube.

1000 can be expressed using the first three primes: 23 x 53.

The 147th prime (853) plus 147 equals 1000.

2999 + 1239 is the smallest 1000-bit prime.

1000 is 2626 in base 7. It is 1331 in base 9.

1000 has two representations as a sum of two squares: 1000 = 102 + 302 = 182 + 262.

T-1000 was the shape-changing robot from the 1991 film Terminator 2: Judgment Day.

Source: Prime Curios!

## Friday, December 14, 2012

### 1672

1672 = 2 x 2 x 2 x 11 x 19.

1672 is 767 in base 15.

1672 is a number that can be expressed as the difference of the squares of primes in just one distinct way (A090781).

1672 is a multiple of 8 that has a digit sum that is a multiple of 8 (A216997).

On an 8 x 8 chessboard, there are 1672 3-step knight's tours summed over all choices for the starting position (A186441).

1672 is the number of obtuse isosceles triangles on a 10 x 10 grid (geoboard) (A190318).

Source: On-Line Encyclopedia of Integer Sequences

## Thursday, December 13, 2012

### 797

797 is a prime number.

797 is a left- and right-trunctable prime number that yields the primes 7, 79, and 97.

797 is the smallest number and the smallest prime n such that googol - n is prime.

797 is the number of functional graphs on 9 vertices.

797 has a representation as a sum of two squares: 797 = 112 + 262.

797 is the hypotenuse of a primitive Pythagorean triple: 7972 = 5552 + 5722.

797 is 665 in base 11 and 565 in base 12. It is 494 in base 13.

Source: Number Gossip

## Wednesday, December 12, 2012

### 395

395 = 5 x 79.

395 is the sum of three consecutive primes (A034961): 395 = 127 + 131 + 137. It is also the sum of five consecutive primes (A034964): 395 = 71 + 73 + 79 + 83 + 89.

395 is the smallest multiple of 5 with a digit sum of 17 (A069534).

395 is the smallest sum of 5 consecutive odd primes that is a multiple of 5 (A132810).

395 is the sum of 15 but no fewer nonzero fourth powers (A046046).

Source: On-Line Encyclopedia of Integer Sequences

## Tuesday, December 11, 2012

### 533

533 = 13 x 41.

533 is the number of degree sequences for graphs with five vertices.

533 has two representations as a sum of two squares: 533 = 22 + 232 = 72 + 222.

533 is the hypotenuse of two primitive Pythagorean triples: 5332 = 922 + 5252 = 3082 + 4352.

533 is 201202 in base 4. It is 191 in base 19.

533 is the sum of three consecutive primes: 533 = 173 + 179 + 181. It is also the sum of five consecutive primes: 533 = 101 + 103 + 107 + 109 + 113.

## Monday, December 10, 2012

### 885

885 = 3 x 5 x 59.

885 is a divisor of 584 - 1.

885!/588! - 1 is prime.

885 is an enneagonal pyramidal number (A007584).

885 is 31311 in base 4 and 311 in base 17. It is 531 in base 13.

Source: Prime Curios!

## Friday, December 7, 2012

### 595

595 = 5 x 7 x 17.

595 is the 34th triangular number (the sum of the first 34 integers).

595 is a centered nonagonal number.

595 is the number of ways to tile a 3 x 18 rectangle with 3 x 1 rectangles.

595 is 151 in base 22.

## Thursday, December 6, 2012

### 996

996 = 2 x 2 x 3 x 83.

996 has a square (992016) formed by inserting a block of digits inside itself (A045953, A046829), A046851, and A086457).

The sum of the squares of the first 996 primes is a prime (A098561).

996 is the number of cubic lattice walks that start and end at the origin after 6 steps, not touching the origin at intermediate stages (A049037).

## Wednesday, December 5, 2012

### 436

436 = 2 x 2 x 109.

436 has a representation as a sum of two squares: 436 = 62 + 202.

436 is the maximum number of regions into which 29 lines divide a plane (A000124).

436 is the smallest number whose cube (82881856) contains four 8s.

436 is 232 in base 14.

436 is the number of ways to cover (without overlapping) a ring lattice (necklace) of 29 sites with molecules that are 8 sites wide (A058365).

## Tuesday, December 4, 2012

### 586

586 = 2 x 293.

586 is the smallest number that appears in its factorial six times.

586 is 4321 in base 5.

586 is the sum of three distinct positive cubes (A024975): 586 = 73 + 63 + 33.

586 has a representation as a sum of two squares: 586 = 152 + 192.

586 is a truncated octahedral number.

## Monday, December 3, 2012

### 684

684 = 2 x 2 x 3 x 3 x 19.

684 has the representation 684 = 36 - 45.

684 is the sum of three consecutive cubes (A024975): 684 = 53 + 63 + 73.

684 is 221100 in base 3.

684 is the largest number that is 38 times the sum of its digits (A052489): 684 = 38 x 18.

Halley's Comet was visible from Earth in the year 684.