• Source: 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), the perimeter of a Pythagorean triangle (75 + 308 + 317) 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, nontotient, Harshad number
703 = 19 × 37, triangular number, hexagonal number, smallest number requiring 73 fifth powers for Waring representation, Kaprekar number, 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 (sequence A000124 in the OEIS), area code for the Charlotte, NC area.
705 = 3 × 5 × 47, sphenic number, smallest Bruckman-Lucas pseudoprime (sequence A005845 in the OEIS)
706 = 2 × 353, nontotient, Smith number
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).
708 = 22 × 3 × 59, number of partitions of 28 that do not contain 1 as a part
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
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.
715 = 5 × 11 × 13, sphenic number, pentagonal number, pentatope number ( binomial coefficient







(


13
4


)






{\displaystyle {\tbinom {13}{4}}}

), 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), Sophie Germain prime, safe prime, sum of seven consecutive primes (89 + 97 + 101 + 103 + 107 + 109 + 113), Chen prime, Eisenstein prime with no imaginary part




= 720s

=

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, 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, area of a square with diagonal 38
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
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, the number of n-queens problem solutions for n = 10,
725 = 52 × 29, side length of an almost-equilateral Heronian triangle
726 = 2 × 3 × 112, pentagonal pyramidal number
727 = prime number, palindromic prime, lucky prime,
728 = 23 × 7 × 13, nontotient, Smith number, cabtaxi number, 728!! - 1 is prime, 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 because of these properties, a perfect totient number.
centered octagonal number, 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
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 {1, 2, 3, 4} that are closed under union and intersection
733 = prime number, emirp, balanced prime, permutable prime, sum of five consecutive primes (137 + 139 + 149 + 151 + 157)
734 = 2 × 367, nontotient, number of traceable graphs on 7 nodes
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, 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, lucky prime, happy number, prime index prime




= 740s

=
740 = 22 × 5 × 37, nontotient, number of connected squarefree graphs on 9 nodes
741 = 3 × 13 × 19, sphenic number, triangular number
742 = 2 × 7 × 53, sphenic number, decagonal number, icosahedral number. It is the smallest number that is one more than triple its reverse. Lazy caterer number (sequence A000124 in the OEIS). 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
746 = 2 × 373 = 15 + 24 + 36 = 17 + 24 + 36, nontotient, number of non-normal semi-magic squares with sum of entries equal to 6
747 = 32 × 83 =




⌊



4

23



3

23




⌋



{\displaystyle \left\lfloor {\frac {4^{23}}{3^{23}}}\right\rfloor }

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




= 750s

=
750 = 2 × 3 × 53, enneagonal number.
751 = prime number, Chen prime, emirp
752 = 24 × 47, nontotient, number of partitions of 11 into parts of 2 kinds
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
755 = 5 × 151, number of vertices in a regular drawing of the complete bipartite graph K9,9.
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
759 = 3 × 11 × 23, sphenic number, sum of five consecutive primes (139 + 149 + 151 + 157 + 163), a q-Fibonacci number for q=3




= 760s

=
760 = 23 × 5 × 19, centered triangular number, number of fixed heptominoes.
761 = prime number, emirp, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, centered square number
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
764 = 22 × 191, telephone number
765 = 32 × 5 × 17, octagonal pyramidal number
a Japanese word-play for Namco;
766 = 2 × 383, centered pentagonal number, 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, sum of eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109)
769 = prime number, Chen prime, lucky prime, Proth prime




= 770s

=
770 = 2 × 5 × 7 × 11, nontotient, Harshad number





∑

n
=
0


10



770

n




{\displaystyle \sum _{n=0}^{10}770^{n}}

is prime
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
⁡

(



2
π

771


)



{\displaystyle \cos \left({\frac {2\pi }{771}}\right)}

can be written in terms of square roots.
772 = 22 × 193, 772!!!!!!+1 is prime
773 = prime number, Eisenstein prime with no imaginary part, tetranacci number, 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
776 = 23 × 97, refactorable number, number of compositions of 6 whose parts equal to q can be of q2 kinds

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.
778 = 2 × 389, nontotient, Smith number
779 = 19 × 41, highly cototient number




= 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 (sequence A000124 in the OEIS)
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




{\displaystyle 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

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
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)
792 = 23 × 32 × 11, number of integer partitions of 21, binomial coefficient







(


12
5


)






{\displaystyle {\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, nontotient
795 = 3 × 5 × 53, sphenic number, Mertens function(795) = 0, number of permutations of length 7 with 2 consecutive ascending pairs
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!
799 = 17 × 47, smallest number with digit sum 25




References

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