600 (number)

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.

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^[2]
• 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
• 607 – prime number, sum of three consecutive primes (197 + 199 + 211), Mertens function(607) = 0, balanced prime,^[3] strictly non-palindromic number,^[4] 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^[5]
• 609 = 3 × 7 × 29, sphenic number, strobogrammatic number^[6]

610s

• 610 = 2 × 5 × 61, sphenic number, Fibonacci number,^[7] Markov number,^[8] 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 (sequence A007602 in the OEIS), 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.^[9]
  • 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,^[10] 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,^[11] prime index prime, index of prime Lucas number^[9]
  • 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,^[12] alternating factorial^[13]

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^[14]
• 621 = 3³ × 23, Harshad number, the discriminant of a totally real cubic field^[15]
• 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^[16]
• 624 = 2⁴ × 3 × 13 = J₄(5),^[17] sum of a twin prime (311 + 313), Harshad number, Zuckerman number
• 625 = 25² = 5⁴, sum of seven consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103), centered octagonal number,^[18] 1-automorphic number, Friedman number since 625 = 5⁶⁻²,^[19] 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,^[20] Smith number^[21]
• 628 = 2² × 157, nontotient, totient sum for first 45 integers
• 629 = 17 × 37, highly cototient number,^[22] Harshad number, number of diagonals in a 37-gon^[23]

630s

• 630 = 2 × 3² × 5 × 7, sum of six consecutive primes (97 + 101 + 103 + 107 + 109 + 113), triangular number, hexagonal number,^[24] sparsely totient number,^[25] Harshad number, balanced number^[26]
• 631 = Cuban prime number, centered triangular number,^[27] centered hexagonal number,^[28] Chen prime, lazy caterer number (sequence A000124 in the OEIS)
• 632 = 2³ × 79, refactorable number, number of 13-bead necklaces with 2 colors^[29]
• 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^[21]
• 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^[30]
  • "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,^[21] Mertens function(636) = 0
• 637 = 7² × 13, Mertens function(637) = 0, decagonal number^[31]
• 638 = 2 × 11 × 29, sphenic number, sum of four consecutive primes (151 + 157 + 163 + 167), nontotient, centered heptagonal number^[32]
• 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,^[33] number of 1's in all partitions of 24 into odd parts,^[34] number of acres in a square mile
• 641 = prime number, Sophie Germain prime,^[35] factor of 4294967297 (the smallest nonprime Fermat number), Chen prime, Eisenstein prime with no imaginary part, Proth prime^[36]
• 642 = 2 × 3 × 107 = 1⁴ + 2⁴ + 5⁴,^[37] sphenic number, admirable number
• 643 = prime number, largest prime factor of 123456
• 644 = 2² × 7 × 23, nontotient, Perrin number,^[38] Harshad number, common umask, admirable number
• 645 = 3 × 5 × 43, sphenic number, octagonal number, Smith number,^[21] Fermat pseudoprime to base 2,^[39] 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^[40]
• 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^[41]
• 648 = 2³ × 3⁴ = A331452(7, 1),^[5] Harshad number, Achilles number, area of a square with diagonal 36^[42]
• 649 = 11 × 59, Blum integer

650s

• 650 = 2 × 5² × 13, primitive abundant number,^[43] square pyramidal number,^[44] pronic number,^[1] 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,^[45] nonagonal number^[46]
• 652 = 2² × 163, maximal number of regions by drawing 26 circles^[47]
• 653 = prime number, Sophie Germain prime,^[35] balanced prime,^[3] Chen prime, Eisenstein prime with no imaginary part
• 654 = 2 × 3 × 109, sphenic number, nontotient, Smith number,^[21] admirable number
• 655 = 5 × 131, number of toothpicks after 20 stages in a three-dimensional grid^[48]
• 656 = 2⁴ × 41 = ${\displaystyle \lfloor {\frac {3^{16}}{2^{16}}}\rfloor }$,^[49] in Judaism, 656 is the number of times that Jerusalem is mentioned in the Hebrew Bible or Old Testament
• 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,^[35] 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,^[22] Eisenstein prime with no imaginary part, strictly non-palindromic number^[4]

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^[25]
  • Sum of 11th row when writing the natural numbers as a triangle.^[50]
  • Harshad 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 ${\displaystyle 5n^{2}-5n+1}$
  • Hexagram number of the form ${\displaystyle 6n^{2}-6n+1}$ i.e. a star number
• 662 = 2 × 331, nontotient, member of Mian–Chowla sequence^[51]
• 663 = 3 × 13 × 17, sphenic number, Smith number^[21]
• 664 = 2³ × 83, refactorable number, number of knapsack partitions of 33^[52]
  • 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^[23]
• 666 = 2 × 3² × 37, Harshad number, repdigit
• 667 = 23 × 29, lazy caterer number (sequence A000124 in the OEIS)
• 668 = 2² × 167, nontotient
• 669 = 3 × 223, blum integer

670s

• 670 = 2 × 5 × 67, sphenic number, octahedral number,^[53] 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,^[54] Zuckerman number, admirable number
• 673 = prime number, Proth prime^[36]
• 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^[55]
• 678 = 2 × 3 × 113, sphenic number, nontotient, number of surface points of an octahedron with side length 13,^[56] 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^[57]

680s

• 680 = 2³ × 5 × 17, tetrahedral number,^[58] nontotient
• 681 = 3 × 227, centered pentagonal number^[2]
• 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^[59]
• 683 = prime number, Sophie Germain prime,^[35] sum of five consecutive primes (127 + 131 + 137 + 139 + 149), Chen prime, Eisenstein prime with no imaginary part, Wagstaff prime^[60]
• 684 = 2² × 3² × 19, Harshad number, number of graphical forest partitions of 32^[61]
• 685 = 5 × 137, centered square number^[62]
• 686 = 2 × 7³, nontotient, number of multigraphs on infinite set of nodes with 7 edges^[63]
• 687 = 3 × 229, 687 days to orbit the Sun (Mars) D-number^[64]
• 688 = 2⁴ × 43, Friedman number since 688 = 8 × 86,^[19] 2-automorphic number^[65]
• 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^[66]

690s

• 690 = 2 × 3 × 5 × 23, sum of six consecutive primes (103 + 107 + 109 + 113 + 127 + 131), sparsely totient number,^[25] Smith number,^[21] 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^[67]
• 693 = 3² × 7 × 11, triangular matchstick number,^[68] the number of sections in Ludwig Wittgenstein's Philosophical Investigations.
• 694 = 2 × 347, centered triangular number,^[27] nontotient, smallest pandigital number in base 5.^[69]
• 695 = 5 × 139, 695!! + 2 is prime.^[70]
• 696 = 2³ × 3 × 29, 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^[71]
• 697 = 17 × 41, cake number; the number of sides of Colorado^[72]
• 698 = 2 × 349, nontotient, sum of squares of two primes^[73]
• 699 = 3 × 233, D-number^[64]

References