700 (number) explained

Number:	700
Lang1:	Armenian
Lang1 Symbol:	Չ
Lang2:	Hebrew
Lang2 Symbol:	ת"ש / ן
Lang3:	Babylonian cuneiform
Lang4:	Egyptian hieroglyph
Lang4 Symbol:

700 (seven hundred) is the natural number following 699 and preceding 701.

It is the sum of four consecutive primes (167 + 173 + 179 + 181), the perimeter of a Pythagorean triangle (75 + 308 + 317)^[1] and a Harshad number.

Integers from 701 to 799

Nearly all of the palindromic integers between 700 and 800 (i.e. nearly all numbers in this range that have both the hundreds and units digit be 7) are used as model numbers for Boeing Commercial Airplanes.

700s

• 701 = prime number, sum of three consecutive primes (229 + 233 + 239), Chen prime, Eisenstein prime with no imaginary part
• 702 = 2 × 3³ × 13, pronic number,^[2] nontotient, Harshad number
• 703 = 19 × 37, triangular number,^[3] hexagonal number,^[4] smallest number requiring 73 fifth powers for Waring representation, Kaprekar number,^[5] area code for Northern Virginia along with 571, a number commonly found in the formula for body mass index
• 704 = 2⁶ × 11, Harshad number, lazy caterer number, area code for the Charlotte, NC area.
  • 705 = 3 × 5 × 47, sphenic number, smallest Bruckman-Lucas pseudoprime 706 = 2 × 353, nontotient, Smith number^[6]
• 707 = 7 × 101, sum of five consecutive primes (131 + 137 + 139 + 149 + 151), palindromic number, number of lattice paths from (0,0) to (5,5) with steps (0,1), (1,0) and, when on the diagonal, (1,1).^[7]
• 708 = 2² × 3 × 59, number of partitions of 28 that do not contain 1 as a part^[8]
• 709 = prime number; happy number. It is the seventh in the series 2, 3, 5, 11, 31, 127, 709 where each number is the nth prime with n being the number preceding it in the series, therefore, it is a prime index number.

710s

• 710 = 2 × 5 × 71, sphenic number, nontotient, number of forests with 11 vertices ^{[9] [10]}
• 711 = 3² × 79, Harshad number, number of planar Berge perfect graphs on 7 nodes. Also the phone number of Telecommunications Relay Service, commonly used by the deaf and hard-of-hearing.
• 712 = 2³ × 89, refactorable number, sum of the first twenty-one primes, totient sum for first 48 integers. It is the largest known number such that it and its 8th power (66,045,000,696,445,844,586,496) have no common digits.
• 713 = 23 × 31, Blum integer, main area code for Houston, TX. In Judaism there are 713 letters on a Mezuzah scroll.
• 714 = 2 × 3 × 7 × 17, sum of twelve consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83), nontotient, balanced number, member of Ruth–Aaron pair (either definition); area code for Orange County, California.
  • Flight 714 to Sidney is a Tintin graphic novel.
  • 714 is the badge number of Sergeant Joe Friday.

\tbinom { 13 } 4 ),^[12] Harshad number, member of Ruth-Aaron pair (either definition)

• The product of 714 and 715 is the product of the first 7 prime numbers (2, 3, 5, 7, 11, 13, and 17)
• 716 = 2² × 179, area code for Buffalo, NY
• 717 = 3 × 239, palindromic number
• 718 = 2 × 359, area code for Brooklyn, NY and Bronx, NY
• 719 = prime number, factorial prime (6! − 1),^[13] Sophie Germain prime,^[14] safe prime,^[15] sum of seven consecutive primes (89 + 97 + 101 + 103 + 107 + 109 + 113), Chen prime, Eisenstein prime with no imaginary part

720s

See main article: 720 (number).

• 720 = 2⁴ × 3² × 5.
  • 6 factorial, highly composite number, Harshad number in every base from binary to decimal, highly totient number.
  • two round angles (= 2 × 360).
  • five gross (= 500 duodecimal, 5 × 144).
  • 241-gonal number.
• 721 = 7 × 103, sum of nine consecutive primes (61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101), centered hexagonal number,^[16] smallest number that is the difference of two positive cubes in two ways,
• 722 = 2 × 19², nontotient, number of odd parts in all partitions of 15,^[17] area of a square with diagonal 38^[18]
  • G.722 is a freely available file format for audio file compression. The files are often named with the extension "722".
• 723 = 3 × 241, side length of an almost-equilateral Heronian triangle^[19]
• 724 = 2² × 181, sum of four consecutive primes (173 + 179 + 181 + 191), sum of six consecutive primes (107 + 109 + 113 + 127 + 131 + 137), nontotient, side length of an almost-equilateral Heronian triangle,^[20] the number of n-queens problem solutions for n = 10,
• 725 = 5² × 29, side length of an almost-equilateral Heronian triangle^[21]
• 726 = 2 × 3 × 11², pentagonal pyramidal number^[22]
• 727 = prime number, palindromic prime, lucky prime,^[23]
• 728 = 2³ × 7 × 13, nontotient, Smith number, cabtaxi number,^[24] 728!! - 1 is prime,^[25] number of cubes of edge length 1 required to make a hollow cube of edge length 12, 728⁶⁴ + 1 is prime, number of connected graphs on 5 labelled vertices
• 729 = 27² = 9³ = 3⁶.
  • the square of 27, and the cube of 9, the sixth power of three, and as a consequence of these properties, a perfect totient number.^[26]
  • centered octagonal number,^[27] Smith number
  • the number of times a philosopher's pleasure is greater than a tyrant's pleasure according to Plato in the Republic
  • the largest three-digit cube. (9 x 9 x 9)
  • the only three-digit sixth power. (3 x 3 x 3 x 3 x 3 x 3)

730s

• 730 = 2 × 5 × 73, sphenic number, nontotient, Harshad number, number of generalized weak orders on 5 points ^[28]
• 731 = 17 × 43, sum of three consecutive primes (239 + 241 + 251), number of Euler trees with total weight 7
• 732 = 2² × 3 × 61, sum of eight consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103 + 107), sum of ten consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97), Harshad number, number of collections of subsets of that are closed under union and intersection
• 733 = prime number, emirp, balanced prime,^[29] permutable prime, sum of five consecutive primes (137 + 139 + 149 + 151 + 157)
• 734 = 2 × 367, nontotient, number of traceable graphs on 7 nodes ^[30]
• 735 = 3 × 5 × 7², Harshad number, Zuckerman number, smallest number such that uses same digits as its distinct prime factors
• 736 = 2⁵ × 23, centered heptagonal number,^[31] happy number, nice Friedman number since 736 = 7 + 3⁶, Harshad number
• 737 = 11 × 67, palindromic number, blum integer.
• 738 = 2 × 3² × 41, Harshad number.
• 739 = prime number, strictly non-palindromic number,^[32] lucky prime, happy number, prime index prime

740s

• 740 = 2² × 5 × 37, nontotient, number of connected squarefree graphs on 9 nodes ^[33]
• 741 = 3 × 13 × 19, sphenic number, triangular number
• 742 = 2 × 7 × 53, sphenic number, decagonal number,^[34] icosahedral number. It is the smallest number that is one more than triple its reverse. Lazy caterer number . Number of partitions of 30 into divisors of 30.
• 743 = prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part
• 744 = 2³ × 3 × 31, sum of four consecutive primes (179 + 181 + 191 + 193). It is the coefficient of the first degree term of the expansion of Klein's j-invariant, and the zeroth degree term of the Laurent series of the J-invariant. Furthermore, 744 = 3 × 248 where 248 is the dimension of the Lie algebra E₈.
• 745 = 5 × 149 = 2⁴ + 3⁶, number of non-connected simple labeled graphs covering 6 vertices^[35]
• 746 = 2 × 373 = 1⁵ + 2⁴ + 3⁶ = 1⁷ + 2⁴ + 3⁶, nontotient, number of non-normal semi-magic squares with sum of entries equal to 6^[36]
• 747 = 3² × 83 =

\left\lfloor { (4 ^{23})/(3 ^{23}) } \right\rfloor,^[37] palindromic number.

• 748 = 2² × 11 × 17, nontotient, happy number, primitive abundant number^[38]
• 749 = 7 × 107, sum of three consecutive primes (241 + 251 + 257), blum integer

750s

• 750 = 2 × 3 × 5³, enneagonal number.^[39]
• 751 = prime number, Chen prime, emirp
• 752 = 2⁴ × 47, nontotient, number of partitions of 11 into parts of 2 kinds^[40]
• 753 = 3 × 251, blum integer
• 754 = 2 × 13 × 29, sphenic number, nontotient, totient sum for first 49 integers, number of different ways to divide a 10 × 10 square into sub-squares ^[41]
• 755 = 5 × 151, number of vertices in a regular drawing of the complete bipartite graph K_9,9.
• 756 = 2² × 3³ × 7, sum of six consecutive primes (109 + 113 + 127 + 131 + 137 + 139), pronic number, Harshad number
• 757 = prime number, palindromic prime, sum of seven consecutive primes (97 + 101 + 103 + 107 + 109 + 113 + 127), happy number.
  • "The 757" is a local nickname for the Hampton Roads area in the U.S. state of Virginia, derived from the telephone area code that covers almost all of the metropolitan area
• 758 = 2 × 379, nontotient, prime number of measurement ^[42]
• 759 = 3 × 11 × 23, sphenic number, sum of five consecutive primes (139 + 149 + 151 + 157 + 163), a q-Fibonacci number for q=3 ^[43]

760s

• 760 = 2³ × 5 × 19, centered triangular number,^[44] number of fixed heptominoes.
• 761 = prime number, emirp, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, centered square number^[45]
• 762 = 2 × 3 × 127, sphenic number, sum of four consecutive primes (181 + 191 + 193 + 197), nontotient, Smith number, admirable number, number of 1's in all partitions of 25 into odd parts, see also Six nines in pi
• 763 = 7 × 109, sum of nine consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103), number of degree-8 permutations of order exactly 2 ^[46]
• 764 = 2² × 191, telephone number^[47]
• 765 = 3² × 5 × 17, octagonal pyramidal number ^[48]
  • a Japanese word-play for Namco;
• 766 = 2 × 383, centered pentagonal number,^[49] nontotient, sum of twelve consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89)
• 767 = 13 × 59, Thabit number (2⁸ × 3 − 1), palindromic number.
• 768 = 2⁸ × 3,^[50] sum of eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109)
• 769 = prime number, Chen prime, lucky prime, Proth prime^[51]

770s

• 770 = 2 × 5 × 7 × 11, nontotient, Harshad number
  • $\backslash sum\_^\; 770^$ is prime^[52]
  • Famous room party in New Orleans hotel room 770, giving the name to a well known science fiction fanzine called File 770
  • Holds special importance in the Chabad-Lubavitch Hasidic movement.
• 771 = 3 × 257, sum of three consecutive primes in arithmetic progression (251 + 257 + 263). Since 771 is the product of the distinct Fermat primes 3 and 257, a regular polygon with 771 sides can be constructed using compass and straightedge, and

\cos\left( (2 \pi)/(771) \right) can be written in terms of square roots.

• 772 = 2² × 193, 772!!!!!!+1 is prime^[53]
• 773 = prime number, Eisenstein prime with no imaginary part, tetranacci number,^[54] prime index prime, sum of the number of cells that make up the convex, regular 4-polytopes
• 774 = 2 × 3² × 43, nontotient, totient sum for first 50 integers, Harshad number
• 775 = 5² × 31, member of the Mian–Chowla sequence^[55]
• 776 = 2³ × 97, refactorable number, number of compositions of 6 whose parts equal to q can be of q² kinds

See main article: 777 (number).

• 777 = 3 × 7 × 37, sphenic number, Harshad number, palindromic number, 3333 in senary (base 6) counting.
  • The numbers 3 and 7 are considered both "perfect numbers" under Hebrew tradition.^{[56] [57]}
• 778 = 2 × 389, nontotient, Smith number
• 779 = 19 × 41, highly cototient number^[58]

780s

• 780 = 2² × 3 × 5 × 13, sum of four consecutive primes in a quadruplet (191, 193, 197, and 199); sum of ten consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101), triangular number, hexagonal number, Harshad number
  • 780 and 990 are the fourth smallest pair of triangular numbers whose sum and difference (1770 and 210) are also triangular.
• 781 = 11 × 71. 781 is the sum of powers of 5/repdigit in base 5 (11111), Mertens function(781) = 0, lazy caterer number
• 782 = 2 × 17 × 23, sphenic number, nontotient, pentagonal number, Harshad number, also, 782 gear used by U.S. Marines
• 783 = 3³ × 29, heptagonal number
• 784 = 2⁴ × 7² = 28² =

1 ^{3+2} ^{3+3} ^{3+4} ^{3+5} ^{3+6} ^{3+7} ^{3} , the sum of the cubes of the first seven positive integers, happy number

• 785 = 5 × 157, Mertens function(785) = 0, number of series-reduced planted trees with 6 leaves of 2 colors ^[59]

See main article: 786 (number).

• 786 = 2 × 3 × 131, sphenic number, admirable number. See also its use in Muslim numerological symbolism.
• 787 = prime number, sum of five consecutive primes (149 + 151 + 157 + 163 + 167), Chen prime, lucky prime, palindromic prime.
• 788 = 2² × 197, nontotient, number of compositions of 12 into parts with distinct multiplicities ^[60]
• 789 = 3 × 263, sum of three consecutive primes (257 + 263 + 269), Blum integer

790s

• 790 = 2 × 5 × 79, sphenic number, nontotient, a Harshad number in bases 2, 7, 14 and 16, an aspiring number, the aliquot sum of 1574.
• 791 = 7 × 113, centered tetrahedral number, sum of the first twenty-two primes, sum of seven consecutive primes (101 + 103 + 107 + 109 + 113 + 127 + 131)

\tbinom { 12 } 5 , Harshad number, sum of the nontriangular numbers between successive triangular numbers

• 793 = 13 × 61, Mertens function(793) = 0, star number, happy number
• 794 = 2 × 397 = 1⁶ + 2⁶ + 3⁶,^[62] nontotient
• 795 = 3 × 5 × 53, sphenic number, Mertens function(795) = 0, number of permutations of length 7 with 2 consecutive ascending pairs ^[63]
• 796 = 2² × 199, sum of six consecutive primes (113 + 127 + 131 + 137 + 139 + 149), Mertens function(796) = 0
• 797 = prime number, Chen prime, Eisenstein prime with no imaginary part, palindromic prime, two-sided prime, prime index prime.
• 798 = 2 × 3 × 7 × 19, Mertens function(798) = 0, nontotient, product of primes indexed by the prime exponents of 10! ^[64]
• 799 = 17 × 47, smallest number with digit sum 25 ^[65]

Notes and References

1. 2022-05-31.
2. Web site: Sloane's A002378 : Oblong (or promic, pronic, or heteromecic) numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
3. Web site: Sloane's A000217 : Triangular numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
4. Web site: Sloane's A000384 : Hexagonal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
5. Web site: Sloane's A006886 : Kaprekar numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
6. Web site: Sloane's A006753 : Smith numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
7. 2022-05-22.
8. 2022-06-02.
9. Classes of perfect graphs - ScienceDirect. Discrete Mathematics . Creation and Recreation: A Tribute to the Memory of Claude Berge . 6 October 2006 . 306 . 19 . 2529–2571 . 10.1016/j.disc.2006.05.021 . Hougardy . Stefan . free .
10. 2022-05-22.
11. Web site: Sloane's A000326 : Pentagonal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
12. Web site: Sloane's A000332 : Binomial coefficient binomial(n,4). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
13. Web site: Sloane's A088054 : Factorial primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
14. Web site: Sloane's A005384 : Sophie Germain primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
15. Web site: Sloane's A005385 : Safe primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
16. Web site: Sloane's A003215 : Hex (or centered hexagonal) numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
17. 2022-05-22.
18. 2*n^2.
19. 2022-05-22.
20. 4*a(n-1) - a(n-2) with a(0) = 2, a(1) = 4. 2022-05-22.
21. 2022-05-22.
22. Web site: Sloane's A002411 : Pentagonal pyramidal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
23. Web site: Sloane's A031157 : Numbers that are both lucky and prime. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
24. Web site: Sloane's A047696 : Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
25. 2022-05-24.
26. Web site: Sloane's A082897 : Perfect totient numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
27. Web site: Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
28. 2022-05-22.
29. Web site: Sloane's A006562 : Balanced primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
30. 2022-05-22.
31. Web site: Sloane's A069099 : Centered heptagonal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
32. Web site: Sloane's A016038 : Strictly non-palindromic numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
33. 2022-05-23.
34. Web site: Sloane's A001107 : 10-gonal (or decagonal) numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
35. 2022-05-23.
36. 2022-05-30.
37. 2022-05-30.
38. Web site: Sloane's A091191 : Primitive abundant numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
39. Web site: Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
40. Product_ 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds. 2022-05-30.
41. 2022-05-23.
42. 2022-05-23.
43. 3. 2022-05-23.
44. Web site: Sloane's A005448 : Centered triangular numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
45. Web site: Sloane's A001844 : Centered square numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
46. 2022-05-23.
47. Web site: Sloane's A000085 : Number of self-inverse permutations on n letters, also known as involutions. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
48. 2022-05-23.
49. Web site: Sloane's A005891 : Centered pentagonal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
50. 3*2^n. 2022-05-30.
51. Web site: Sloane's A080076 : Proth primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
52. 2022-05-30.
53. 2022-05-30.
54. Web site: Sloane's A000078 : Tetranacci numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
55. Web site: Sloane's A005282 : Mian-Chowla sequence. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
56. Web site: Posner . Eliezer . On the Meaning of Three . Chabad . 2 July 2016.
57. Web site: Dennis . Geoffrey . Judaism & Numbers . My Jewish Learning . 2 July 2016.
58. Web site: Sloane's A100827 : Highly cototient numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-06-11.
59. 2022-05-24.
60. 2022-05-24.
61. a(n) = number of partitions of n.
62. a(n) = 1^n + 2^n + 3^n.
63. 2022-05-24.
64. 2022-05-24.
65. 2022-05-24.