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), and is a Harshad number.

Integers from 701 to 799[]

Nearly all of the palindromic integers between 700 and 800 are used as model numbers for Boeing Commercial Airplanes, and the only one not officially used by Boeing, 797, is commonly speculated to be the number of the next new Boeing commercial airplane.^[1]

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, area code for the Charlotte, NC area.
• 705 = 3 × 5 × 47, sphenic number, smallest Lucas pseudoprime
• 706 = 2 × 353, nontotient, Smith number^[6]
• 707 = 7 × 101, sum of five consecutive primes (131 + 137 + 139 + 149 + 151), palindromic number, model number for the Boeing 707
• 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); 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,^[7] pentatope number ( binomial coefficient ${\displaystyle {\tbinom {13}{4}}}$ ),^[8]

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, model number for the Boeing 717
• 718 = 2 × 359, area code for Brooklyn, NY and Bronx, NY
• 719 = prime number, factorial prime (6! − 1),^[9] Sophie Germain prime,^[10] safe prime,^[11] 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,^[12] 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^[13]
• 727 = prime number, palindromic prime, lucky prime,^[14]
  • model number for the Boeing 727
  • known in popular culture in the rhythm game osu! and related with the song Blue Zenith by xi in the album Parousia.
• 728 = 2³ × 7 × 13, nontotient, Smith number,^[6] cabtaxi number^[15]
• 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.^[16]
  • centered octagonal number,^[17] Smith number^[6]
  • 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, 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,^[18] 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,^[19] 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,^[20] lucky prime,^[14] happy number

740s[]

• 740 = 2² × 5 × 37, nontotient
• 741 = 3 × 13 × 19, sphenic number, triangular number^[3]
• 742 = 2 × 7 × 53, sphenic number, decagonal number.^[21] 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^[22]
• 749 = 7 × 107, sum of three consecutive primes (241 + 251 + 257)

750s[]

• 750 = 2 × 3 × 5³, enneagonal number.^[23]
• 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,^[2] Harshad number
• 757 = prime number, palindromic prime, sum of seven consecutive primes (97 + 101 + 103 + 107 + 109 + 113 + 127), happy number, model number for the Boeing 757
  • "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^[24]
• 761 = prime number, emirp, Sophie Germain prime,^[10] Chen prime, Eisenstein prime with no imaginary part, centered square number^[25]
• 762 = 2 × 3 × 127, sphenic number, sum of four consecutive primes (181 + 191 + 193 + 197), nontotient, Smith number,^[6] 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^[26]
• 765 = 3² × 5 × 17
  • a Japanese word-play for Namco;
• 766 = 2 × 383, centered pentagonal number,^[27] 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, model number for the Boeing 767
• 768 = 2⁸ × 3, sum of eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109)
• 769 = prime number, Chen prime, lucky prime,^[14] Proth prime^[28]

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^[29]
• 774 = 2 × 3² × 43, nontotient, totient sum for first 50 integers, Harshad number
• 775 = 5² × 31, member of the Mian–Chowla sequence,^[30] 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.^[31][32]
• 778 = 2 × 389, nontotient, Smith number^[6]
• 779 = 19 × 41, highly cototient number^[33]

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,^[3] hexagonal number,^[4] 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,^[7] 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,^[14] palindromic prime, model number for the Boeing 787 Dreamliner
• 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,^[34] binomial coefficient ${\displaystyle {\tbinom {12}{5}}}$, Harshad number
• 793 = 13 × 61, Mertens function(793) = 0, star number,^[35] 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, two-sided prime, speculated model number for the Boeing New Midsize Airplane
• 798 = 2 × 3 × 7 × 19, Mertens function(798) = 0, nontotient
• 799 = 17 × 47

References[]