# 600 (number)

(Redirected from 691 (number))
 ← 599 600 601 →
Cardinalsix hundred
Ordinal600th
(six hundredth)
Factorization23 × 3 × 52
Divisors1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600
Greek numeralΧ´
Roman numeralDC
Binary10010110002
Ternary2110203
Octal11308
Duodecimal42012

600 (six hundred) is the natural number following 599 and preceding 601.

## Mathematical properties

Six hundred is a composite number, an abundant number, a pronic number[1] and a Harshad number.

## Integers from 601 to 699

### 610s

• 613 = prime number, first number of prime triple (p, p + 4, p + 6), middle number of sexy prime triple (p − 6, p, p + 6). Geometrical numbers: Centered square number with 18 per side, circular number of 21 with a square grid and 27 using a triangular grid. Also 17-gonal. Hypotenuse of a right triangle with integral sides, these being 35 and 612. Partitioning: 613 partitions of 47 into non-factor primes, 613 non-squashing partitions into distinct parts of the number 54. Squares: Sum of squares of two consecutive integers, 17 and 18. Additional properties: a lucky number.
• 614 = 2 × 307, nontotient. According to Rabbi Emil Fackenheim, the number of Commandments in Judaism should be 614 rather than the traditional 613.
• 615 = 3 × 5 × 41, sphenic number

### 620s

• 620 = 22 × 5 × 31, sum of four consecutive primes (149 + 151 + 157 + 163), sum of eight consecutive primes (61 + 67 + 71 + 73 + 79 + 83 + 89 + 97). The sum of the first 620 primes is itself prime.[10]
• 621 = 33 × 23, Harshad number
• 622 = 2 × 311, nontotient. It is also the standard diameter of modern road bicycle wheels (622 mm, from hook bead to hook bead)
• 623 = 7 × 89
• 624 = 24 × 3 × 13, sum of a twin prime (311 + 313), Harshad number, Zuckerman number
• 625 = 54 = 252, sum of seven consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103), centered octagonal number,[11] 1-automorphic number, Friedman number since 625 = 56−2[12]
• 626 = 2 × 313, nontotient
• 627 = 3 × 11 × 19, sphenic number, number of integer partitions of 20,[13] Smith number[14]
• 628 = 22 × 157, nontotient, totient sum for first 45 integers
• 629 = 17 × 37, highly cototient number,[15] Harshad number

### 640s

• 640 = 27 × 5, Harshad number, number of acres in a square mile
• 641= prime number, Sophie Germain prime,[22] factor of 4294967297 (the smallest nonprime Fermat number), Chen prime, Eisenstein prime with no imaginary part, Proth prime[23]
• 642 = 2 × 3 × 107, sphenic number
• 643 = prime number, largest prime factor of 123456
• 644 = 22 × 7 × 23, nontotient, Perrin number,[24] Harshad number, common umask.
• 645 = 3 × 5 × 43, sphenic number, Smith number,[14] Fermat pseudoprime to base 2,[25] Harshad number
• 646 = 2 × 17 × 19, sphenic number, also ISO 646 is the ISO's standard for international 7-bit variants of ASCII
• 647 = prime number, sum of five consecutive primes (113 + 127 + 131 + 137 + 139), Chen prime, Eisenstein prime with no imaginary part
• 648 = 23 × 34, Harshad number
• 649 = 11 × 59

### 660s

• 660 = 22 × 3 × 5 × 11, sum of four consecutive primes (157 + 163 + 167 + 173), sum of six consecutive primes (101 + 103 + 107 + 109 + 113 + 127), sum of eight consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97 + 101), sparsely totient number,[17] Harshad number
• 661 = prime number, sum of three consecutive primes (211 + 223 + 227), Mertens function sets new low of −11 which stands until 665, star number
• 662 = 2 × 331, nontotient, member of Mian–Chowla sequence[30]
• 663 = 3 × 13 × 17, sphenic number, Smith number[14]
• 664 = 23 × 83
• 665 = 5 × 7 × 19, sphenic number, Mertens function sets new low of −12 which stands until 1105

### 680s

• 680 = 23 × 5 × 17, tetrahedral number,[33] nontotient
• 681 = 3 × 227, centered pentagonal number[3]
• 682 = 2 × 11 × 31, sphenic number, sum of four consecutive primes (163 + 167 + 173 + 179), sum of ten consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89)
• 683 = prime number, Sophie Germain prime,[22] sum of five consecutive primes (127 + 131 + 137 + 139 + 149), Chen prime, Eisenstein prime with no imaginary part, Wagstaff prime[34]
• 684 = 22 × 32 × 19, Harshad number
• 685 = 5 × 137, centered square number[35]
• 686 = 2 × 73, nontotient
• 687 = 3 × 229
• 688 = 24 × 43, Friedman number since 688 = 8 × 86[12]
• 689 = 13 × 53, sum of three consecutive primes (227 + 229 + 233), sum of seven consecutive primes (83 + 89 + 97 + 101 + 103 + 107 + 109). Strobogrammatic number[36]

### 690s

• 690 = 2 × 3 × 5 × 23, sum of six consecutive primes (103 + 107 + 109 + 113 + 127 + 131), sparsely totient number,[17] Smith number,[14] Harshad number
• ISO 690 is the ISO's standard for bibliographic references
• 691 = prime number, (negative) numerator of the Bernoulli number B12 = -691/2730. Ramanujan's tau function τ and the divisor function σ11 are related by the remarkable congruence τ(n) ≡ σ11(n) (mod 691).
• In number theory, 691 is a "marker" (similar to the radioactive markers in biology): whenever it appears in a computation, one can be sure that Bernoulli numbers are involved.
• 692 = 22 × 173
• 693 = 32 × 7 × 11, the number of the "non-existing" Alabama State Constitution amendment, the number of sections in Ludwig Wittgenstein's Philosophical Investigations.
• 694 = 2 × 347, centered triangular number,[18] nontotient
• 695 = 5 × 139. The number of people dead in the single deadliest tornado in United States history
• 696 = 23 × 3 × 29, sum of eight consecutive primes (71 + 73 + 79 + 83 + 89 + 97 + 101 + 103), totient sum for first 47 integers
• 697 = 17 × 41; the number of sides of Colorado[37]
• 698 = 2 × 349, nontotient
• 699 = 3 × 233

## References

1. ^ a b "Sloane's A002378 : Oblong (or promic, pronic, or heteromecic) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
2. ^ Lewis and Short, A Latin Dictionary, s.v. sescenti
3. ^ a b "Sloane's A005891 : Centered pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
4. ^ a b "Sloane's A006562 : Balanced primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
5. ^ a b "Sloane's A016038 : Strictly non-palindromic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
6. ^ "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
7. ^ "Sloane's A002559 : Markoff (or Markov) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
8. ^ "Sloane's A007597 : Strobogrammatic primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
9. ^ "Sloane's A005165 : Alternating factorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
10. ^
11. ^ "Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
12. ^ a b "Sloane's A036057 : Friedman numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
13. ^ "Sloane's A000041 : a(n) = number of partitions of n". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
14. "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
15. ^ a b "Sloane's A100827 : Highly cototient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
16. ^ "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
17. ^ a b c "Sloane's A036913 : Sparsely totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
18. ^ a b "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
19. ^ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
20. ^ "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
21. ^ "Sloane's A069099 : Centered heptagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
22. ^ a b c d "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
23. ^ a b "Sloane's A080076 : Proth primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
24. ^ "Sloane's A001608 : Perrin sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
25. ^ "Sloane's A001567 : Fermat pseudoprimes to base 2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
26. ^ "Sloane's A071395 : Primitive abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
27. ^ "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
28. ^ "Sloane's A000326 : Pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
29. ^ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
30. ^ "Sloane's A005282 : Mian-Chowla sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
31. ^ "Sloane's A005900 : Octahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
32. ^ "Sloane's A001599 : Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
33. ^ "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
34. ^ "Sloane's A000979 : Wagstaff primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
35. ^ "Sloane's A001844 : Centered square numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
36. ^ "Sloane's A000787 : Strobogrammatic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
37. ^ [1]