700 (number)

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

It is the sum of four consecutive primes (167 + 173 + 179 + 181). It is a Harshad number.

Integers from 701 to 799

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,^[1] nontotient, Harshad number
• 703 = 19 × 37, triangular number,^[2] hexagonal number,^[3] smallest number requiring 73 fifth powers for Waring representation, Kaprekar number,^[4] area code for Northern Virginia along with 571, a number commonly found in the formula for body mass index
• 704 = 2⁶ × 11, Harshad number, area code for the Charlotte, NC area.
• 705 = 3 × 5 × 47, sphenic number, smallest Lucas pseudoprime
• 706 = 2 × 353, nontotient, Smith number^[5]
• 707 = 7 × 101, sum of five consecutive primes (131 + 137 + 139 + 149 + 151), palindromic number
• 708 = 2² × 3 × 59
• 709 = prime number; happy number.

710s

• 710 = 2 × 5 × 71, sphenic number, nontotient
• 711 = 3² × 79, Harshad number. Also the phone number of Telecommunications Relay Service, commonly used by the deaf and hard-of-hearing.
• 712 = 2³ × 89, 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, main area code for Houston, TX. In Judaism there is 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, member of Ruth–Aaron pair (either definition); the smallest number that uses the same digits in bases 2 and 5, area code for Orange County, California.
  • 714 is the number of career home runs hit by Babe Ruth, a record that stood from his last home run on May 25, 1935 until being broken by Hank Aaron on April 8, 1974.
  • Flight 714 to Sidney is a Tintin graphic novel.
  • 714 is the badge number of Sergeant Joe Friday.
• 715 = 5 × 11 × 13, sphenic number, pentagonal number,^[6] pentatope number ( binomial coefficient ${\displaystyle {\tbinom {13}{4}}}$ ),^[7]

Harshad number, member of Ruth-Aaron pair (either definition)

• 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),^[8] Sophie Germain prime,^[9] safe prime,^[10] sum of seven consecutive primes (89 + 97 + 101 + 103 + 107 + 109 + 113), Chen prime, Eisenstein prime with no imaginary part

720s

• 720 (seven hundred [and] twenty)= 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,^[11] smallest number that is the difference of two positive cubes in two ways,
• 722 = 2 × 19², nontotient
  • G.722 is a freely available file format for audio file compression. The files are often named with the extension "722".
• 723 = 3 × 241
• 724 = 2² × 181, sum of four consecutive primes (173 + 179 + 181 + 191), sum of six consecutive primes (107 + 109 + 113 + 127 + 131 + 137), nontotient
  • the number of n-queens problem solutions for n = 10,
• 725 = 5² × 29
• 726 = 2 × 3 × 11², pentagonal pyramidal number^[12]
• 727 = prime number, palindromic prime, lucky prime^[13]
• 728 = 2³ × 7 × 13, nontotient, Smith number,^[5] cabtaxi number^[14]
• 729 = 3⁶ = 27².
  • the square of 27, and the cube of 9, and as a consequence of these properties, a perfect totient number.^[15]
  • centered octagonal number,^[16] Smith number^[5]
  • 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 largest 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, happy number
• 731 = 17 × 43, sum of three consecutive primes (239 + 241 + 251)
• 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
• 733 = prime number, balanced prime,^[17] permutable prime, sum of five consecutive primes (137 + 139 + 149 + 151 + 157)
• 734 = 2 × 367, nontotient
• 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,^[18] nice Friedman number since 736 = 7 + 3⁶, Harshad number
• 737 = 11 × 67, palindromic number, model number of the Boeing 737 jet airliner.
• 738 = 2 × 3² × 41, Harshad number, designation for a Boeing 737-800 jet airliner.
• 739 = prime number, strictly non-palindromic number,^[19] lucky prime,^[13] happy number

740s

• 740 = 2² × 5 × 37, nontotient
• 741 = 3 × 13 × 19, sphenic number, triangular number^[2]
• 742 = 2 × 7 × 53, sphenic number, decagonal number.^[20] It is the smallest number that is one more than triple its reverse.

• 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. Furthermore, 744 =3 × 248 where 248 is the dimension of the Lie algebra E₈.
• 745 = 5 × 149
• 746 = 2 × 373, nontotient
  • 746 = 1⁷ + 2⁴ + 3⁶
• 747 = 3² × 83, palindromic number, model number of the Boeing 747 jet airliner
• 748 = 2² × 11 × 17, nontotient, happy number, primitive abundant number^[21]
• 749 = 7 × 107, sum of three consecutive primes (241 + 251 + 257)

750s

• 750 = 2 × 3 × 5³, enneagonal number.^[22]
• 751 = prime number, Chen prime
• 752 = 2⁴ × 47, nontotient
• 753 = 3 × 251
• 754 = 2 × 13 × 29, sphenic number, nontotient, totient sum for first 49 integers
• 755 = 5 × 151. In 1976, Major League Baseball player Hank Aaron ended his career with a Major League record 755 home runs (record now held by Barry Bonds).
• 756 = 2² × 3³ × 7, sum of six consecutive primes (109 + 113 + 127 + 131 + 137 + 139), pronic number,^[1] 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
• 759 = 3 × 11 × 23, sphenic number, sum of five consecutive primes (139 + 149 + 151 + 157 + 163)

760s

• 760 = 2³ × 5 × 19, centered triangular number^[23]
• 761 = prime number, Sophie Germain prime,^[9] Chen prime, Eisenstein prime with no imaginary part, centered square number^[24]
• 762 = 2 × 3 × 127, sphenic number, sum of four consecutive primes (181 + 191 + 193 + 197), nontotient, Smith number,^[5] see also Six nines in pi
• 763 = 7 × 109, sum of nine consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103)
• 764 = 2² × 191, telephone number^[25]
• 765 = 3² × 5 × 17
  • a Japanese word-play for Namco;
• 766 = 2 × 383, centered pentagonal number,^[26] nontotient, sum of twelve consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89), happy number
• 767 = 13 × 59, Thabit number (2⁸ × 3 − 1), palindromic number
• 768 = 2⁸ × 3, sum of eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109)
• 769 = prime number, Chen prime, lucky prime,^[13] Proth prime^[27]

770s

• 770 = 2 × 5 × 7 × 11, nontotient, Harshad number
  • 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 ${\displaystyle \cos \left({\frac {2\pi }{771}}\right)}$ can be written in terms of square roots.
• 772 = 2² × 193
• 773 = prime number, Eisenstein prime with no imaginary part, tetranacci number^[28]
• 774 = 2 × 3² × 43, nontotient, totient sum for first 50 integers, Harshad number
• 775 = 5² × 31, member of the Mian–Chowla sequence,^[29] happy number
• 776 = 2³ × 97

• 777 = 3 × 7 × 37, sphenic number, Harshad number, palindromic number, model number of the Boeing 777 jet airliner, 3333 in senary (base 6) counting.
  • The numbers 3 and 7 are considered both "perfect numbers" under Hebrew tradition.^[30][31]
• 778 = 2 × 389, nontotient, Smith number^[5]
• 779 = 19 × 41, highly cototient number^[32]

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,^[2] hexagonal number,^[3] 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, sum of powers of 5/repdigit in base 5 (11111), Mertens function(781) = 0
• 782 = 2 × 17 × 23, sphenic number, nontotient, pentagonal number,^[6] Harshad number, also, 782 gear used by U.S. Marines
• 783 = 3³ × 29
• 784 = 2⁴ × 7² = 28² = ${\displaystyle 1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}+7^{3}}$, the sum of the cubes of the first seven integers, happy number
• 785 = 5 × 157, Mertens function(785) = 0

• 786 = 2 × 3 × 131, sphenic 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,^[13] palindromic prime.
• 788 = 2² × 197, nontotient
• 789 = 3 × 263, sum of three consecutive primes (257 + 263 + 269)

790s

• 790 = 2 × 5 × 79, sphenic number, nontotient
• 791 = 7 × 113, sum of the first twenty-two primes, sum of seven consecutive primes (101 + 103 + 107 + 109 + 113 + 127 + 131)
• 792 = 2³ × 3² × 11, number of partitions of 21,^[33] binomial coefficient ${\displaystyle {\tbinom {12}{5}}}$, Harshad number
• 793 = 13 × 61, Mertens function(793) = 0, star number,^[34] happy number
• 794 = 2 × 397, nontotient
• 795 = 3 × 5 × 53, Mertens function(795) = 0
• 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
• 798 = 2 × 3 × 7 × 19, Mertens function(798) = 0, nontotient
• 799 = 17 × 47

References