## 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.

## Friday, November 30, 2012

### 499

499 is a prime number.

499 is the smallest prime that is the sum of a prime number of first odd primes: 499 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61.

499499 ends with the digits 499499. In general, its odd powers end with 499 and its even powers end with 001.

Separate 499 into two (49, 9) and three (4, 9, 9) groups and get two primes after replacing each subset with each subset with its square root, namely 73 and 233. Replacing each digit of these latter two primes with their cubes results in two new primes: 34327 and 82727.

499 is the number of ways to place 26 points on a 13 x 13 grid so that no three points are on a line.

499 is the smallest number with the property that its first 12 multiples contain the digit 9.

499 seconds is the average time interval for sunlight from the sun to reach Earth.

Source: Prime Curios!

## Thursday, November 29, 2012

### 1507

1507 = 11 x 137.

1507 is a divisor of 1004 - 1 (A027890). It is also a divisor of 1016 - 1 (A111211).

1507 is the number of partitions of 32 that do not contain 1 as a part (A002865).

1507 is the sum of exactly three sets of Fibonacci numbers (A122195).

The sums of the digits of 1507 in base 2 (10111100011), base 3 (2001211), and base 5 (22012) are equal (A135121).

The word "America" was first used on a map published in the year 1507.

## Wednesday, November 28, 2012

### 1422

1422 = 2 x 3 x 3 x 79.

1422 is the number of ways to stack 27 boxes in a line so that each box lies on the table or on a box next to two boxes (A001522).

1422 is a divisor of 553 - 1.

14222 = 2022084 has only even digits (A030097).

The sum of the digits of 1422 and 14222 are both squares (A197125): 1 + 4 + 2 + 2 = 9 = 32 and 2 + 0 + 2 + 0 + 8 + 4 = 16 = 42.

## Tuesday, November 27, 2012

### 394

394 = 2 x 197.

394 has a representation as a sum of two squares: 394 = 132 + 152.

394 is a Schroeder number (A006318).

394 is 202 in base 14.

The sum of the digits of 394 equals the sum of the digits of the 394th prime (2707) (A033549).

394 is the middle semiprime of three consecutive semiprimes: 393, 394, 395.

## Monday, November 26, 2012

### 487

487 is a prime number.

487 is the smallest prime p such that p and p3 (115501303) have the same digit sum: 4 + 8 + 7 = 19 = 1 + 1 + 5 + 5 + 0 + 1 + 3 + 0 + 3.

487 is the number of Hadamard matrices of order 28.

487 is 200001 in base 3. It is 13213 in base 4 and 2131 in base 6. It is 747 in base 8 and 191 in base 18.

487 is the sum of three consecutive primes: 487 = 157 + 163 + 167.

Source: Number Gossip

## Wednesday, November 21, 2012

### 371

371 = 7 x 53.

371 is a narcissistic number (Armstrong number): 371 = 33 + 73 + 13.

371 is the largest known semiprime that is also the sum of consecutive prime numbers from 7 (smallest prime factor) to 53 (largest prime factor): 371 = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53.

371 is the sum of three consecutive primes and seven consecutive primes: 371 = 113 + 117 + 131 = 41 + 43 + 47 + 53 + 59 + 61 + 67.

371 is the sum of five positive fifth powers: 371 = 35 + 25 + 25 + 25 + 25.

Source: Number Gossip

## Tuesday, November 20, 2012

### 561

561 = 3 x 11 x 17.

561 is 1000110001 in base 2 (binary) (A006995).

561 is the 33rd triangular number (A000217), the sum of the first 33 integers. It is also a hexagonal number (A000384) and a dodecagonal number (A051624).

561 is the smallest Carmichael number (A002997).

561 is a centered icosahedral number (cuboctahedral number) (A005902).

Source: Number Gossip

## Monday, November 19, 2012

### 952

952 = 2 x 2 x 2 x 7 x 17.

952 = 93 + 53 + 23 + (9 x 5 x 2).

952 is 4224 in base 6.

952 is the number of regions in an 14-gon with all diagonals drawn (A007678).

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

952 is the number of possible queen moves on a 7 x 7 chessboard (A035005).

952 Caia is an asteroid orbiting the sun.

## Friday, November 16, 2012

### 454

454 = 2 x 227.

454 is the largest number known that cannot be written as a sum of seven or fewer cubes (A018889).

454 is a Smith number (A006753). The sum of its digits (4 + 5 + 4 = 13) is equal to the sum of the digits of its prime factors (2 + 2 + 2 + 7 = 13).

1 + 8 + 10 + 17 = 36 = 2 + 5 + 13 + 16
12 + 82 + 102 + 172 = 454 = 22 + 52 + 132 + 162
13 + 83 + 103 + 173 = 6426 = 23 + 53 + 133 + 163

454 is 383 in base 11 (A029966) and 246 in base 14.

## Thursday, November 15, 2012

### 2034

2034 = 2 x 3 x 3 x 113.

2034 is the number of self-avoiding walks of length 9.

2034 is the number of ways of writing 4 as a sum of nine squares (A008452).

2034 has a representation as a sum of two squares: 2034 = 32 + 452.

2034 is a divisor of 3516 - 1.

2034 is 909 in base 15. It is 369 in base 25.

## Wednesday, November 14, 2012

### 2006

2006 = 2 x 17 x 59.

2006 is 2202022 in base 3. It is 434 in base 22.

2006 is a term in the sequence starting with 1, then adding 35, then repeatedly reversing the digits and adding 35 (A118632).

2006 is a multiple of 17 such that its reversal minus 1 (6001) is also a multiple of 17 (A166398).

2006 is the number of line segments connecting exactly 10 points in a 26 x 26 grid (A177726).

A transit of Mercury as seen from Earth took place in the year 2006 (A171466).

Source: On-Line Encyclopedia of Integer Sequences

## Tuesday, November 13, 2012

### 1203

1203 = 3 x 401.

1203 is a divisor of 204 - 1.

1203 is the smallest number n for which the concatenation n, (n + 1), (n + 2), . . . . (n + 34) is prime (A052079).

1203 is 102303 in base 4, 14303 in base 5, and 303 in base 20. It is 717 in base 13.

12032 = 1447209 and 12033 = 1740992427 have the same set of digits (A029797 and A045620).

## Monday, November 12, 2012

### 1010

1010 = 2 x 5 x 101.

1010 has two representations as a sum of two squares: 1010 = 72 + 312 = 132 + 292.

1010 is the sum of two distinct powers of 10 (A052216 and A038444): 1010 = 103 + 101.

1010 is the number of ways to tile a 5 x 12 rectangle with pentominoes.

1010 is 2642 in base 7 and 1342 in base 9. It is 262 in base 21.

1010 is the number of primes less than 104 that do not contain the digit 0 (A091634).

## Friday, November 9, 2012

### 902

902 = 2 x 11 x41. It is the smallest number beginning with 9 that is the product of exactly three primes (A106419).

902 is a value of n such that n(n + 7) is palindromic: 902(909) = 819918.

902 is 1020102 in base 3, 12102 in base 5, and 4102 in base 6. It is 1212 in base 9 and 321 in base 17. It is 545 in base 13 and 252 in base 20.

902, 903, and 904 have the same number of divisors (A005238).

902 is a multiple of 11 such that sum of its digits is also a multiple of 11 (A216995).

## Thursday, November 8, 2012

### 1483

1483 is a prime number.

1481 and 1483 form a twin prime pair.

1483 is the smallest prime that is the sum of three prime cubes: 1483 = 33 + 53 + 113.

1483 has the representation 2 x 36 + 25.

1483 is a divisor of 383 - 1.

The Petronas Towers in Kuala Lumpur, Malaysia, are 1483 feet tall.

Source: Prime Curios!

## Wednesday, November 7, 2012

### 4656

4656 = 2 x 2 x 2 x 2 x 3 x 97.

4656 is the 96th triangular number, with digit sum 21 (A068131).

4656 is 1020300 in base 4 and 1230 in base 16.

4656 is a structured pentagonal hexacontahedral number (A100169).

4656 is the number of 3 x 3 arrays of squares on integers summing to 18 (A159401).

Source: Numeropedia

## Tuesday, November 6, 2012

### 442

442 = 2 x 13 x 17.

442 has two representations as a sum of two squares: 442 = 12 + 212 = 92 + 192.

442 is the smallest composite number n that divides F(n - 1) where F(k) are the Fibonacci numbers.

442 is the number of planar partitions of 13.

442 is the sum of eight consecutive primes: 442 = 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71.

442 is 3232 in base 5 and 101 in base 21.

Source: Number Gossip

## Monday, November 5, 2012

### 693

693 = 3 x 3 x 7 x 11.

693 is 1010110101 in base 2 (binary). It is 3113 in base 6, 414 in base 13, and 313 in base 15.

693 has the representation 693 = 36 - 36.

A primitive Pythagorean triangles of sides (693, 1924, 2045) has area 666,666.

693 is the number of sections in Ludwig Wittgenstein's Philosophical Investigations.

Source: Numeropedia

## Friday, November 2, 2012

### 884

884 = 2 x 2 x 13 x 17.

884 has two representations as a sum of two squares: 884 = 102 + 282 = 202 + 222.

The fifth root of 884 has a decimal part that begins with the digits 884.

884 is 202 in base 21.

The sum of the divisors of 884 is a square (A006532): 1 + 2 + 4 + 13 + 17 + 26 + 34 + 52 + 68 + 221 + 442 + 884 = 1764 = 422.

## Thursday, November 1, 2012

### 407

407 = 11 x 37.

407 is a divisor of 1003 - 1.

407 is the maximum number of regions into which 28 lines divide the plane.

407 is the largest number that is the sum of the cubes of its digits: 407 = 43 + 03 + 73. It is a narcissistic number.

407 = (7 - 3)3 + 73.

407 is 1515 in base 6.

407 is the sum of three consecutive primes: 407 = 131 + 137 + 139.

Source: Number Gossip

## Wednesday, October 31, 2012

### 182

182 = 2 x 7 x 13.

182 is a pronic number; it is the product of two consecutive numbers: 182 = 13 x 14.

182 is 222 in base 9 and 77 in base 25. It is 20202 in base 3 and 1212 in base 5.

182 is the number of connected bipartite graphs with 8 vertices.

182 = 33 + 33 + 43 + 43 = 142 - 14.

The USS Salmon  (SS-182) was a United States Navy submarine during World War II.

Source: Number Gossip

## Tuesday, October 30, 2012

### 777

777 = 3 x 7 x 37.

777 is a divisor of 1003 - 1.

777 is 3333 in base 6.

Surrounding the number 777 with the number 11 forms the prime numbers 11777, 77711, 1177711, and 11117771111.

The Boeing 777 is a long-range, wide-body, twin-engine jet airliner.

Source: Numeropedia

## Monday, October 29, 2012

### 869

869 = 11 x 79.

869 is the number of different resistances that can be created in a circuit of nine equal resistors (A048211).

The sum of the digits of 869 equals the sum of the digits of the 869th prime (6737) (A033549). 869 and the 869th prime (6737) have only the digit 6 in common (A107937).

869 is a number that yields a prime when 0 is inserted anywhere in it (but not added to the end or beginning) (A216169).

## Friday, October 26, 2012

### 2142

2142 = 2 x 3 x 3 x 7 x 17.

2142 is a divisor of 554 - 1.

2142 is 32032 in base 5.

2142 is the sum of the totient function for the first 83 integers.

2142 is the number of ways of writing the number 63 as the sum of nine nonnegative cubes (A173682).

2142 is the sum of 10 consecutive primes (A127337): 2142 = 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239.

Source: On-Line Encyclopedia of Integer Sequences

## Thursday, October 25, 2012

### 198

198 = 2 x 3 x 3 x 11.

198 is the smallest number with ten representations as a sum of four positive squares: 198 = 12 + 22 + 72 + 122 = 12 + 42 + 92 + 102 = 22 + 32 + 42 + 132 = 22 + 32 + 82 + 112 = 22 + 52 + 52 + 122 = 22 + 72 + 82 + 92 = 32 + 32 + 62 + 122 = 32 + 52 + 82 + 102 = 42 + 52 + 62 + 112 = 62 + 72 + 72 + 82.

198 is the first number between a twin prime pair such that if multiplied by its reversal (198 x 891 = 176418), the product is also between a twin prime pair.

198 can be formed by concatenating 19 and 8, where 19 is the 8th prime number. 198 is the (19 x 8 = 152)nd composite number.

198 = 11 + 99 + 88 = (100 - 1)(100-98).

198 x 27 = 5346 = 18 x 297. Each equality uses each of the nine digits 1 to 9 once.

Source: Prime Curios!

## Wednesday, October 24, 2012

### 756

756 = 2 x 2 x 3 x 3 x 3 x 7.

756 is a pronic number, the product of two consecutive numbers, 27 and 28.

12 x 63 = 756 = 21 x 36.

756 is the maximum number of regions into which 14 spheres divide space.

756!/657! - 1 is a prime.

756 is 11011 in base 5. It is 3300 in base 6.

756 is the sum of six consecutive primes: 756 = 109 + 113 + 127 + 131 + 137 + 139.

Source: Number Gossip

## Tuesday, October 23, 2012

### 387

387 = 3 x 3 x 43.

387 is the lesser of two consecutive numbers divisible by a square (A068781).

387 is 110000011 in base 2 (binary) (A006995).

387 x 7 - 38 = 2671, the 387th prime.

387 is the smallest number with sort-then-add persistence 10.

Source: Prime Curios!

## Monday, October 22, 2012

### 667

667 = 23 x 29. It is the product of two successive primes (A006094) and two distinct Sophie Germain primes (A157344).

667 is the maximum number of regions into which 36 lines divide the plane (A000124).

667 is the number of asymmetric trees with 16 vertices (A000220).

667, 669, and 671 are all semiprimes (A092125).

667 YCF was on the license plates of the car driven by Meeno Paloma (Lou Diamond Philips) in the 2009 movie The Triangle.

Source: Prime Curios!

## Friday, October 19, 2012

### 355

355 = 5 x 71.

355 divided by the prime number 113 yields a nice approximation of pi: 355/113 = 3.141592. . .

355 is the number of labeled topologies with four elements.

355 = 3 x 5! - 5.

355 is 543 in base 8 and 434 in base 9.

Source: Prime Curios!

## Thursday, October 18, 2012

### 740

740 = 2 x 2 x 5 x 37.

740 has two representations as a sum of two squares: 740 = 82 + 262 = 162 + 222.

740 is the number of self-avoiding walks of length 8.

740 is 3232 in base 8 and 345 in base 15.

740 is a hypotenuse for which there exist four distinct integer triangles (A084648).

## Wednesday, October 17, 2012

### 129

129 = 3 x 43.

129 is 10000001 in base (binary). It is 333 in base 6.

129 is the smallest number with four representations as a sum of three positive squares: 129 = 12 + 82 + 82 = 22 + 22 + 112 = 22 + 52 + 102 = 42 + 72 + 82.

129 is the sum of the first ten primes: 129 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29.

129 is the smallest sum of distinct seventh powers: 129 = 17 + 27.

Source: Number Gossip

## Tuesday, October 16, 2012

### 930

930 = 2 x 3 x 5 x 31. It is a pronic number, the product of two consecutive integers (A002378): 930 = 30 x 31. It is also the product of a prime and the previous number (A036689).

930 is the number of even permutations of seven elements with no fixed points.

930 is 656 in base 12 and 567 in base 13.

The sum of the divisors of 930 is a square (A006532): 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 + 31 + 62 + 93 + 155 + 186 + 310 + 465 + 930 = 2304 = 482.

## Monday, October 15, 2012

### 1124

1124 = 2 x 2 x 281.

1124 has a representation as a sum of two squares: 1124 = 102 + 322.

1124 is the maximum number of regions into which 34 circles divide a plane.

1124 is a number whose product of digits is equal to its sum of digits: 1 + 1 + 2 + 4 = 8 = 1 x 1 x 2 x 4.

1124 is 686 in base 13, 464 in base 16, 323 in base 19, and 272 in base 22.

1124 is a Leyland number (A076980).

Source: On-Line Encyclopedia of Integer Sequences