## Wednesday, July 31, 2013

### 1441

1441 = 11 x 131.

All the divisors of 1441 are palindromic (A062687): 1, 11, 131, and 1441.

1441 is a palindrome with exactly two prime factors (A046328).

1441 is a palindrome in base 10 and in base 6 (10401) (A029963).

1441 is a star number (or centered dodecagonal number) (A003154).

1441 is a concentric hexagonal number (A032528).

1441 is a number n such that 6n + 1, 12n + 1, and 18n + 1 are all primes (A046025).

## Tuesday, July 30, 2013

### 591

591 = 3 x 197.

591 is the number of ways to stack 23 boxes in a line so that each box lies on the table or on a box next to two boxes.

591 is a divisor of 144 - 1.

591 is 303 in base 14.

591 is a number n such that n2 (349281) contains exactly six different digits (A054034).

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

## Monday, July 29, 2013

### 858

858 = 2 x 3 x 11 x 13.

858 is the smallest palindrome with four distinct prime factors. Note that the reversal of each prime factor is itself a prime.

858 is 454 in base 14 (A029969).

858 is a Guiga number (A007850): a composite number n such that p divides n/p - 1 for every prime divisor p of n.

The sum of the squares of the first 858 primes is prime (A098561).

Source: Prime Curios!

## Friday, July 26, 2013

### 674

674 = 2 x 337.

674 is 220222 in base 3 and 22202 in base 4. It is 828 in base 9.

674 is the sum of the cubes of the first six Fibonacci numbers (A005968).

674 is a number n such that n3 is a sum of two successive primes (A074925).

674 has a representation as a sum of two squares: 674 = 72 + 252.

Source: On-Line Encyclopedia of Integer Sequences

## Thursday, July 25, 2013

### 4128

4128 = 2 x 2 x 2 x 2 x 2 x 3 x 43.

4128 is the smallest number with the property that its first 10 multiples contain the digit 2.

4128 is 1000000100000 in base 2 (binary), 1000200 in base 4, 10040 in base 8, and 1020 in base 16. It is 15015 in base 7 and 3113 in base 11.

4128 is a divisor of 496 - 1.

4128 is a number that can be expressed as the difference of the squares of primes in just two distinct ways (A090788).

4128 = 84 + 8 x 4.

## Wednesday, July 24, 2013

### 497

497 = 7 x 71.

497 is the maximum number of regions into which 31 lines divide a plane (A000124).

497 is the number of graphs with 8 edges.

497 is the sum of five consecutive primes: 497 = 89 + 97 + 101 + 103 + 107.

497 is 232 in base 15.

The sum of the divisors of 497 is a square (A006532): 1 + 7 + 71 + 497 = 576 = 242.

497 and its reversal, 794, are distinct semiprimes (A097393).

## Tuesday, July 23, 2013

### 3019

3019 is a prime number. Its reversal, 9103, is also a prime.

The sum of the digits of 3019 (and 9103) is 13, and 13 and its reversal, 31, are both primes (A178092).

3019 is the smallest positive number that can be written as the sum of distinct Fibonacci numbers in 50 ways (A013583).

3019 is a prime of the form 6p + 1, where p (503) is also a prime (A051644).

9103 - 3019 = 6084 = 782, a perfect square (A006567).

Source: On-Line Encyclopedia of Integer Sequences

## Monday, July 22, 2013

### 334

334 = 2 x 167. It is an even semiprime (A100484).

334 x 3! x 3! x 4! + 1 = 288,577, a prime.

334 is the number of trees on on 13 vertices with diameter 7.

334 is 2314 in base 5 and 1314 in base 6.

334 and 335 are consecutive semiprimes (A070552).

The sum of the first 334 primes is a prime (A013916).

The sum of the squares of the prime factors of 334 is a prime (A134616): 22 + 1672 = 27,893, a prime.

Source: Prime Curios!

## Friday, July 19, 2013

### 570

570 = 2 x 3 x 5 x 19. 570 is divisible by exactly four different primes (A033993) Alternatively, it is the product of four distinct primes (A046386).

570 is the product of the only four prime Roman numeral palindromes: 570 = II x III x V x XIX.

570 is a number with 16 divisors (A030634).

570 is the smallest positive even integer that is an unordered sum of two (odd) primes in exactly 31 ways (A023036 and A001172).

570 is the number of 2 x 2 singular integer matrices with elements from {0, . . ., 9} (A059306).

The sum of the digits of 570 raised to its digits' powers is prime (A046431): 55 + 77 + 00 = 826,669, a prime.

Source: Number Gossip

## Thursday, July 18, 2013

### 2002

2002 = 2 x 7 x 11 x 13.

2002 is the sum of 14 consecutive primes: 2002 = 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179.

2002 is a palindrome, as is its square (A057135): 20022 = 4008004.

2002303 + 3 is a 1001-digit titanic prime.

2002 is the smallest multiple of 7 using only the digits 0 and 2 (A078241).

New Year's Day 2002 CE is day number 730853, a prime number.

Source: Prime Curios!

## Wednesday, July 17, 2013

### 882

882 = 2 x 3 x 3 x 7 x 7.

882 is the largest number less than 1000 that is preceded and followed by primes (between twin primes).

882 is the smallest number whose square begins with three 7s: 8822 = 777,924.

882 is 12012 in base 5. It is 616 in base 12 and 242 in base 20.

882 has a representation as a sum of two squares: 882 = 212 + 212.

Source: Prime Curios!

## Tuesday, July 16, 2013

### 1578

1578 = 2 x 3 x 263.

1578 is the number of Hamiltonian paths of a 3 x 8 rectangle graph (A003685).

1578 is the sum of the first 45 composite numbers (A053767).

The product of the digits of the cube of 1578 is also a cube (A067071): 15783 = 3,929,352,552; 3 x 9 x 2 x 9 x 3 x 5 x 2 x 5 x 5 x 2 = 729,000 = 903.

## Monday, July 15, 2013

### 319

319 = 11 x 29.

319 is the largest number whose cube's digits are all different: 3193 = 32461759.

319 is the sum of three consecutive primes: 319 = 103 + 107 + 109.

Among the partitions of n into distinct primes, the one having the maximum product of parts is not necessarily one of those with the maximum number of parts. The smallest case is 319 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 31 + 37 + 41 + 47 + 53 and 319 = 3 + 5 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47. Hence, 319 is the smallest number with the property that the partition with the largest product does not have a maximum number of parts.

319 cannot be represented as a sum of fewer than 19 fourth powers: 319 = 3 x 34 + 4 x 24 + 12 x 14.

319 is a divisor of 124 - 1.

Source: Number Gossip and Prime Curios!

## Friday, July 12, 2013

### 2628

2628 = 2 x 2 x 3 x 3 x 73.

2628 is the 72nd triangular number (the sum of the first 72 positive integers). Note that its external digits, 2 and 8, form a triangular number when concatenated (A077367). It is the least triangular number divisible by the 21st prime (A112456).

2628 is 101001000100 in base 2 (binary). 2628 contains only the digits 0, 1, and 2 in bases 3, 4, and 6.

2628 is a divisor of 656 - 1.

2628 has a representation as a sum of two squares (A073613): 2628 = 182 + 482.

Source: On-Line Encyclopedia of Integer Sequences

## Thursday, July 11, 2013

### 2370

2370 = 2 x 3 x 5 x 79.

2370 is the sum of 12 consecutive primes (A127339): 2370 = 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229.

2370 is a number n such that the sum of the cubes of the digits of n3 is a perfect cube (A164882): 23703 = 13,312,053,000; 13 + 33 + 33 + 13 + 23 + 53 + 33 = 216 = 63.

2370 is a divisor of 2312 - 1.

2370 is palindromic (3223) in base 9 (A032809).

2370 is the number of three-step self-avoiding walks on a 5 x 5 x 5 cube summed over all starting positions (A187162 and A187164).

Source: On-Line Encyclopedia of Integer Sequences

## Wednesday, July 10, 2013

### 597

597 = 3 x 199.

597 to 611 is the largest gap between two consecutive semiprimes (having exactly two prime factors) less than 1000. It is the smallest semiprime such that gap between it and the next semiprime is 14 (A131109).

597 is a value of n for which n!!! + 1 is prime (A037083).

597 is 4342 in base 5 and 212 in base 17. 597 contains only the digits 1 and 2 in base 4 (21111) (A032925) and base 17.

5972 = 356,409 contains exactly six different digits (A054034).

58! begins with the digits 597. 58 is the smallest integer m such that m! begins with 597 (A076219).

Source: Prime Curios!

## Tuesday, July 9, 2013

### 2537

2537 = 43 x 59.

2537 is a semiprime with distinct prime digits (A108632).

2537 is the smallest number that can be expressed as the sum of a prime number and a nonzero square in just 20 different ways (A064283).

2537 is the sum of all the primes raised to their corresponding powers: 2537 = 21 + 32 + 53 + 74 (A087480).

2537 is a composite number such that exactly eight distinct permutations of digits give primes (A163560).

Source: On-Line Encyclopedia of Integer Sequences

## Monday, July 8, 2013

### 421

421 is a prime number.

419 and 421 are twin primes.

421 is a centered square number.

421 is the smallest number formed by the powers of two in logical order from right to left.

421 is 111 in base 20.

421 is the sum of five consecutive prime numbers: 421 = 73 + 79 + 83 + 89 + 97.

421 has a representation as a sum of two squares: 421 = 142 + 152.

421 is the hypotenuse of a primitive Pythagorean triple: 4212 = 292 + 4202.

All integers ever tested in the Collatz problem eventually end in the sequence 4-2-1.

Source: Prime Curios!

## Friday, July 5, 2013

### 651

651 = 3 x 7 x 31.

651 is a pentagonal number and a nonagonal number. It is the smallest number that is both pentagonal and nonagonal.

651 is 10101 in base 5 and 3003 in base 6.

651 is the number of binary trees of height 4.

651 has a fourth power that is the sum of four fourth powers.

101,556 = 372 x 273 = 651 x 156.

Source: Number Gossip

## Wednesday, July 3, 2013

### 2222

2222 = 2 x 11 x 101.

All of the divisors of 2222 are palindromic (A062687).

2222 is the smallest number divisible by a 1-digit prime, a 2-digit prime, and a 3-digit prime.

2222 = 21 + 22 + 23 + 25 + 27 + 211.

The sum of the reciprocals of the digits of 2222 is an integer (A034708).

The cubic powers 23 = 8; 223 = 10,648; 2223 = 10,941,048; 22223 = 10,970,645,048; 222223 = 10,973,607,685,048; and 2222223 = 10,973,903,978,085,048 do not use the digit 2.

Source: Number Gossip

## Tuesday, July 2, 2013

### 1533

1533 = 3 x 7 x 73. The concatenation of the prime factors of 1533 is a palindrome  (A046452): 3773.

1533 is palindromic in base 2 (binary) (A048700): 10111111101.

1533 is a divisor of 643 - 1.

1533 is 21 x the 21st prime (73) (A033286).

1533 is a number n such that 4n - 1, 8n - 1, and 16n - 1 are primes (A101790), as are 2n + 1, 4n + 1, and 6n + 1 (A124408).

Source: On-Line Encyclopedia of Integer Sequences

## Monday, July 1, 2013

### 366

366 = 2 x 3 x 61.

366 is a divisor of 133 - 1.

366 is 222 in base 13.

366 = 82 + 92 + 102 + 112.

366 is a number n that is divisible by every digit of n (A034838).

366 is a number n such that n10 + n9 + n8 + n7 + n6 + n5 + n4 + n3 + n2 + n + 1 is a prime (A162862).

A leap year has 366 days.

Source: On-Line Encyclopedia of Integer Sequences