600 (number) explained

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, a Harshad number and a largely composite number.

Credit and cars

• In the United States, a credit score of 600 or below is considered poor, limiting available credit at a normal interest rate
• NASCAR runs 600 advertised miles in the Coca-Cola 600, its longest race
• The Fiat 600 is a car, the SEAT 600 its Spanish version

Integers from 601 to 699

600s

• 601 = prime number, centered pentagonal number
• 602 = 2 × 7 × 43, nontotient, number of cubes of edge length 1 required to make a hollow cube of edge length 11, area code for Phoenix, AZ along with 480 and 623
• 603 = 3² × 67, Harshad number, Riordan number, area code for New Hampshire
• 604 = 2² × 151, nontotient, totient sum for first 44 integers, area code for southwestern British Columbia (Lower Mainland, Fraser Valley, Sunshine Coast and Sea to Sky)
• 605 = 5 × 11², Harshad number, sum of the nontriangular numbers between the two successive triangular numbers 55 and 66, number of non-isomorphic set-systems of weight 9
• 606 = 2 × 3 × 101, sphenic number, sum of six consecutive primes (89 + 97 + 101 + 103 + 107 + 109), admirable number, One of the numbers associated with Christ - ΧϚʹ - see the Greek numerals Isopsephy and the reason why other numbers siblings with this one are Beast's numbers.
• 607 – prime number, sum of three consecutive primes (197 + 199 + 211), Mertens function(607) = 0, balanced prime, strictly non-palindromic number, Mersenne prime exponent
• 608 = 2⁵ × 19, Mertens function(608) = 0, nontotient, happy number, number of regions formed by drawing the line segments connecting any two of the perimeter points of a 3 times 4 grid of squares^[1]
• 609 = 3 × 7 × 29, sphenic number, strobogrammatic number

610s

• 610 = 2 × 5 × 61, sphenic number, Fibonacci number, Markov number, also a kind of telephone wall socket used in Australia
• 611 = 13 × 47, sum of the three standard board sizes in Go (9² + 13² + 19²), the 611th tribonacci number is prime
• 612 = 2² × 3² × 17, Harshad number, Zuckerman number, untouchable number, area code for Minneapolis, MN
• 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, index of prime Lucas number.
  • In Judaism the number 613 is very significant, as its metaphysics, the Kabbalah, views every complete entity as divisible into 613 parts: 613 parts of every Sefirah; 613 mitzvot, or divine Commandments in the Torah; 613 parts of the human body.
  • The number 613 hangs from the rafters at Madison Square Garden in honor of New York Knicks coach Red Holzman's 613 victories
• 614 = 2 × 307, nontotient, 2-Knödel number. 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
• 616 = 2³ × 7 × 11, Padovan number, balanced number, an alternative value for the Number of the Beast (more commonly accepted to be 666)
• 617 = prime number, sum of five consecutive primes (109 + 113 + 127 + 131 + 137), Chen prime, Eisenstein prime with no imaginary part, number of compositions of 17 into distinct parts,^[2] prime index prime, index of prime Lucas number
  • Area code 617, a telephone area code covering the metropolitan Boston area
• 618 = 2 × 3 × 103, sphenic number, admirable number
• 619 = prime number, strobogrammatic prime, alternating factorial

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
• 621 = 3³ × 23, Harshad number, the discriminant of a totally real cubic field
• 622 = 2 × 311, nontotient, Fine number, Fine's sequence (or Fine numbers): number of relations of valence >= 1 on an n-set; also number of ordered rooted trees with n edges having root of even degree, it is also the standard diameter of modern road bicycle wheels (622 mm, from hook bead to hook bead)
• 623 = 7 × 89, number of partitions of 23 into an even number of parts
• 624 = 2⁴ × 3 × 13 = J₄(5), sum of a twin prime pair (311 + 313), Harshad number, Zuckerman number
• 625 = 25² = 5⁴, sum of seven consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103), centered octagonal number,^[3] 1-automorphic number, Friedman number since 625 = 5^6-2, one of the two three-digit numbers when squared or raised to a higher power that end in the same three digits, the other being 376
• 626 = 2 × 313, nontotient, 2-Knödel number, Stitch's experiment number
• 627 = 3 × 11 × 19, sphenic number, number of integer partitions of 20,^[4] Smith number
• 628 = 2² × 157, nontotient, totient sum for first 45 integers
• 629 = 17 × 37, highly cototient number, Harshad number, number of diagonals in a 37-gon^[5]

630s

• 630 = 2 × 3² × 5 × 7, sum of six consecutive primes (97 + 101 + 103 + 107 + 109 + 113), the 35th triangular number,^[6] a hexagonal number, sparsely totient number, Harshad number, balanced number, largely composite number
• 631 = Cuban prime number, Lucky prime, centered triangular number, centered hexagonal number, Chen prime, lazy caterer number
• 632 = 2³ × 79, refactorable number, number of 13-bead necklaces with 2 colors
• 633 = 3 × 211, sum of three consecutive primes (199 + 211 + 223), Blum integer; also, in the title of the movie 633 Squadron
• 634 = 2 × 317, nontotient, Smith number
• 635 = 5 × 127, sum of nine consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89), Mertens function(635) = 0, number of compositions of 13 into pairwise relatively prime parts^[7]
  • "Project 635", the Irtysh River diversion project in China involving a dam and a canal
• 636 = 2² × 3 × 53, sum of ten consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83), Smith number, Mertens function(636) = 0
• 637 = 7² × 13, Mertens function(637) = 0, decagonal number
• 638 = 2 × 11 × 29, sphenic number, sum of four consecutive primes (151 + 157 + 163 + 167), nontotient, centered heptagonal number
• 639 = 3² × 71, sum of the first twenty primes, also ISO 639 is the ISO's standard for codes for the representation of languages

640s

• 640 = 2⁷ × 5, Harshad number, refactorable number, hexadecagonal number,^[8] number of 1's in all partitions of 24 into odd parts, number of acres in a square mile
• 641 = prime number, Sophie Germain prime, factor of 4294967297 (the smallest nonprime Fermat number), Chen prime, Eisenstein prime with no imaginary part, Proth prime
• 642 = 2 × 3 × 107 = 1⁴ + 2⁴ + 5⁴,^[9] sphenic number, admirable number
• 643 = prime number, largest prime factor of 123456
• 644 = 2² × 7 × 23, nontotient, Perrin number,^[10] Harshad number, common umask, admirable number
• 645 = 3 × 5 × 43, sphenic number, octagonal number, Smith number, Fermat pseudoprime to base 2, Harshad number
• 646 = 2 × 17 × 19, sphenic number, also ISO 646 is the ISO's standard for international 7-bit variants of ASCII, number of permutations of length 7 without rising or falling successions
• 647 = prime number, sum of five consecutive primes (113 + 127 + 131 + 137 + 139), Chen prime, Eisenstein prime with no imaginary part, 3⁶⁴⁷ - 2⁶⁴⁷ is prime
• 648 = 2³ × 3⁴ = A331452(7, 1), Harshad number, Achilles number, area of a square with diagonal 36^[11]
• 649 = 11 × 59, Blum integer

650s

• 650 = 2 × 5² × 13, primitive abundant number, square pyramidal number, pronic number, nontotient, totient sum for first 46 integers; (other fields) the number of seats in the House of Commons of the United Kingdom, admirable number
• 651 = 3 × 7 × 31, sphenic number, pentagonal number, nonagonal number
• 652 = 2² × 163, maximal number of regions by drawing 26 circles^[12]
• 653 = prime number, Sophie Germain prime, balanced prime, Chen prime, Eisenstein prime with no imaginary part
• 654 = 2 × 3 × 109, sphenic number, nontotient, Smith number, admirable number
• 655 = 5 × 131, number of toothpicks after 20 stages in a three-dimensional grid
• 656 = 2⁴ × 41 =

• 657 = 3² × 73, the largest known number not of the form a²+s with s a semiprime
• 658 = 2 × 7 × 47, sphenic number, untouchable number
• 659 = prime number, Sophie Germain prime, sum of seven consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107), Chen prime, Mertens function sets new low of -10 which stands until 661, highly cototient number, Eisenstein prime with no imaginary part, strictly non-palindromic number

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
  • Sum of 11th row when writing the natural numbers as a triangle.^[14]
  • Harshad number.
  • largely composite number
• 661 = prime number
  • Sum of three consecutive primes (211 + 223 + 227)
  • Mertens function sets new low of -11 which stands until 665
  • Pentagram number of the form

• • Hexagram number of the form

i.e. a star number

• 662 = 2 × 331, nontotient, member of Mian–Chowla sequence
• 663 = 3 × 13 × 17, sphenic number, Smith number
• 664 = 2³ × 83, refactorable number, number of knapsack partitions of 33
  • Telephone area code for Montserrat
  • Area code for Tijuana within Mexico
  • Model number for the Amstrad CPC 664 home computer
• 665 = 5 × 7 × 19, sphenic number, Mertens function sets new low of -12 which stands until 1105, number of diagonals in a 38-gon^[5]
• 666 = 2 × 3² × 37, 36th triangular number,^[15] Harshad number, repdigit
• 667 = 23 × 29, lazy caterer number
• 668 = 2² × 167, nontotient
• 669 = 3 × 223, Blum integer

670s

• 670 = 2 × 5 × 67, sphenic number, octahedral number, nontotient
• 671 = 11 × 61. This number is the magic constant of n×n normal magic square and n-queens problem for n = 11.
• 672 = 2⁵ × 3 × 7, harmonic divisor number, Zuckerman number, admirable number, largely composite number, triperfect number
• 673 = prime number, lucky prime, Proth prime
• 674 = 2 × 337, nontotient, 2-Knödel number
• 675 = 3³ × 5², Achilles number
• 676 = 2² × 13² = 26², palindromic square
• 677 = prime number, Chen prime, Eisenstein prime with no imaginary part, number of non-isomorphic self-dual multiset partitions of weight 10
• 678 = 2 × 3 × 113, sphenic number, nontotient, number of surface points of an octahedron with side length 13,^[16] admirable number
• 679 = 7 × 97, sum of three consecutive primes (223 + 227 + 229), sum of nine consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97), smallest number of multiplicative persistence 5^[17]

680s

• 680 = 2³ × 5 × 17, tetrahedral number,^[18] nontotient
• 681 = 3 × 227, centered pentagonal number
• 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), number of moves to solve the Norwegian puzzle strikketoy^[19]
• 683 = prime number, Sophie Germain prime, sum of five consecutive primes (127 + 131 + 137 + 139 + 149), Chen prime, Eisenstein prime with no imaginary part, Wagstaff prime^[20]
• 684 = 2² × 3² × 19, Harshad number, number of graphical forest partitions of 32^[21]
• 685 = 5 × 137, centered square number^[22]
• 686 = 2 × 7³, nontotient, number of multigraphs on infinite set of nodes with 7 edges^[23]
• 687 = 3 × 229, 687 days to orbit the Sun (Mars) D-number^[24]
• 688 = 2⁴ × 43, Friedman number since 688 = 8 × 86, 2-automorphic number^[25]
• 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

690s

• 690 = 2 × 3 × 5 × 23, sum of six consecutive primes (103 + 107 + 109 + 113 + 127 + 131), sparsely totient number, Smith number, 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, number of partitions of 48 into powers of 2^[26]
• 693 = 3² × 7 × 11, triangular matchstick number,^[27] the number of sections in Ludwig Wittgenstein's Philosophical Investigations.
• 694 = 2 × 347, centered triangular number, nontotient, smallest pandigital number in base 5.^[28]
• 695 = 5 × 139, 695!! + 2 is prime.^[29]
• 696 = 2³ × 3 × 29, sum of a twin prime (347 + 349) sum of eight consecutive primes (71 + 73 + 79 + 83 + 89 + 97 + 101 + 103), totient sum for first 47 integers, trails of length 9 on honeycomb lattice^[30]
• 697 = 17 × 41, cake number; the number of sides of Colorado^[31]
• 698 = 2 × 349, nontotient, sum of squares of two primes
• 699 = 3 × 233, D-number^[24]

References

1. Triangle read by rows: T(n,m) (n >= m >= 1) = number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.
2. 2022-05-24.
3. Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.
4. a(n) = number of partitions of n.
5. a(n) = n*(n+3)/2.
6. Web site: A000217 - OEIS . 2024-11-29 . oeis.org.
7. 2022-05-31.
8. 16-gonal (or hexadecagonal) numbers: a(n) = n*(7*n-6).
9. a(n) = 1^n + 2^n + 5^n. 2022-05-31.
10. Web site: Sloane's A001608 : Perrin sequence. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
11. a(n) = 2*n^2.
12. a(n) = n^2 + n + 2.
13. a(n) = floor(3^n / 2^n).
14. a(n) = n*(n+1)*(n+2)/2.
15. Web site: A000217 - OEIS . 2024-11-29 . oeis.org.
16. 2022-05-31.
17. 2022-05-31.
18. 2016-06-11.
19. 2022-05-31.
20. 2016-06-11.
21. a(n) = Sum_ p(k) where p(k) = number of partitions of k (A000041). 2022-05-31.
22. 2016-06-11.
23. 2022-05-31.
24. 3-Knödel numbers or D-numbers: numbers n > 3 such that n divides k^(n-2)-k for all k with gcd(k, n) = 1. 2022-05-31.
25. 2021-09-01.
26. 2022-05-31.
27. Triangular matchstick numbers: a(n) = 3*n*(n+1)/2. 2022-05-31.
28. a(1) = 1; for n > 1, smallest digitally balanced number in base n.
29. 2022-05-31.
30. 2022-05-18.
31. Web site: Colorado is a rectangle? Think again. 23 January 2023 .