758

Integers from 701 to 799

700s

701 is a prime number, a Chen prime, an Eisenstein prime with no imaginary part, and the sum of three consecutive primes (229 + 233 + 239).

702 = 2 × 33 × 13. It is a pronic number, a nontotient, and a Harshad number.

703 = 19 × 37. It is a hexagonal number, a Kaprekar number and the 37th triangular number. It is the smallest number requiring 73 fifth powers for Waring representation.

703 is commonly found in the formula for body mass index.

704 = 26 × 11. It is a Harshad number and a lazy caterer number.

705 = 3 × 5 × 47. It is a sphenic number the smallest Bruckman-Lucas pseudoprime.

706 = 2 × 353. It is a nontotient a Smith number.

707 = 7 × 101. It is a palindromic number and the sum of five consecutive primes (131 + 137 + 139 + 149 + 151). There are 707 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. There are 708 partitions of 28 that do not contain 1 as a part.

709 is a prime number and a 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. It is a sphenic number and a nontotient. There are 710 forests with 11 vertices.

711 = 32 × 79. It is a Harshad number. There are 711 planar Berge perfect graphs on 7 nodes.

712 = 23 × 89. It is a refactorable number, the totient sum for first 48 integers, and the sum of the first twenty-one primes.

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. It is a Blum integer.

In Judaism there are 713 letters on a Mezuzah scroll.

714 = 2 × 3 × 7 × 17. It is a nontotient and a balanced number, and the sum of twelve consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83). It forms a Ruth–Aaron pair with 715 (either definition). It is the sum of twelve consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83).

The product of 714 and 715 is the product of the first 7 prime numbers (2, 3, 5, 7, 11, 13, and 17).

715 = 5 × 11 × 13. It is a sphenic number, a pentagonal number, and a Harshad number. It forms a Ruth-Aaron pair with 714 (either definition).

It is a pentatope number because 713=

(

13

4

)

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

.

720s

721 = 7 × 103. It is a centered hexagonal number and the sum of nine consecutive primes (61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101). It is the smallest number that is the difference of two positive cubes in two ways.

722 = 2 × 192. It is a nontotient. There are 722 odd parts in all partitions of 15.

723 = 3 × 241. It is the side length of an almost-equilateral Heronian triangle.

724 = 22 × 181. It is a nontotient and the side length of an almost-equilateral Heronian triangle. It is the sum of four consecutive primes (173 + 179 + 181 + 191) and the sum of six consecutive primes (107 + 109 + 113 + 127 + 131 + 137). There are 724 n-queens problem solutions for n = 10.

725 = 52 × 29. It is the side length of an almost-equilateral Heronian triangle.

726 = 2 × 3 × 112. It is a pentagonal pyramidal number.

727 is a prime number, a palindromic prime, and a lucky prime.

728 = 23 × 7 × 13. It is a nontotient, a Smith number, and a cabtaxi number. There are 728 cubes of edge length 1 required to make a hollow cube of edge length 12.There are 728 connected graphs on 5 labelled vertices.

728!! - 1 is prime.

72864 + 1 is prime.

729 = 272 = 93 = 36. It is a perfect totient number, a Smith number, and a centered octagonal number. It is the largest three-digit cube (93) and the only three-digit sixth power (36).

A philosopher king's pleasure is 729 times a tyrant's pleasure according to Plato in the Republic.

730s

730 = 2 × 5 × 73. It is a sphenic number, a nontotient, and a Harshad number. There are 730 generalized weak orders on 5 points.

731 = 17 × 43. It is the sum of three consecutive primes (239 + 241 + 251). There are 731 Euler trees with total weight 7.

732 = 22 × 3 × 61. It is a Harshad number.

It is the sum of eight consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103 + 107) and the sum of ten consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97).

There are 732 collections of subsets of {1, 2, 3, 4} that are closed under union and intersection.

733 is a prime number, a balanced prime, a permutable prime, and an emirp. It is the sum of five consecutive primes (137 + 139 + 149 + 151 + 157)

734 = 2 × 367. It is a nontotient. There are 734 traceable graphs on 7 nodes.

735 = 3 × 5 × 72. It is a Harshad number and a Zuckerman number.

the smallest number such that uses the same digits as its distinct prime factors

736 = 25 × 23. It is a centered heptagonal number, a happy number, a Harshad number, and a nice Friedman number since 736 = 7 + 36.

737 = 11 × 67. It is a palindromic number and a blum integer.

738 = 2 × 32 × 41. It is a Harshad number.

739 is a prime number, a lucky prime, a prime index prime, a happy number, and a strictly non-palindromic number.

740s

740 = 22 × 5 × 37. It is a nontotient. There are 740 connected square free graphs on 9 nodes.

741 = 3 × 13 × 19. It is a sphenic number and the 38th triangular number.

742 = 2 × 7 × 53. It is a sphenic number, a decagonal number, an icosahedral number, and a lazy caterer number. There are 742 partitions of 30 into divisors of 30.

the smallest number that is one more than triple its reverse.

It is a prime number and an eisenstein prime with no imaginary part. 743 is a Sophie Germain prime because 2 × 743 + 1 = 1487 and 1487 is also prime. 743 is an emirp, because 347 (the reversal of its digits) is prime.

There are exactly 743 independent sets in a four-dimensional (16 vertex) hypercube graph, and exactly 743 connected cubic graphs with 16 vertices and girth four.

744 is a semiperfect number and an abundant number.

The j-invariant, an important function in the study of modular forms and Monstrous moonshine, can be written as a Fourier series in which the constant term is 744:

j

(

τ

)

=

q

−

1

+

744

+

196

750s

750 = 2 × 3 × 53. It is an enneagonal number.

751 is a prime number, a Chen prime, and an emirp.

752 = 24 × 47. It is a nontotient. There are 752 partitions of 11 into parts of 2 kinds

753 = 3 × 251. It is a blum integer.

754 = 2 × 13 × 29. It is a sphenic number, a nontotient, and the totient sum for first 49 integers.

There are 754 different ways to divide a 10 × 10 square into sub-squares.

755 = 5 × 151. There are 755 vertices in a regular drawing of the complete bipartite graph K9,9.

756 = 22 × 33 × 7. It is a pronic number, a Harshad number, and the sum of six consecutive primes (109 + 113 + 127 + 131 + 137 + 139).

757 is a prime number, a palindromic prime, a happy number, and the sum of seven consecutive primes (97 + 101 + 103 + 107 + 109 + 113 + 127).

758 = 2 × 379.

It is a nontotient and a prime number of measurement.

759 = 3 × 11 × 23.

It is a sphenic number, a q-Fibonacci number for q=3, and the sum of five consecutive primes (139 + 149 + 151 + 157 + 163).

760s

760 = 23 × 5 × 19.

It is a centered triangular number. There are 760 fixed heptominoes.

761 is a prime number, an emirp, a Sophie Germain prime, a Chen prime, an Eisenstein prime with no imaginary part, and a centered square number.

762 = 2 × 3 × 127. It is a sphenic number, a nontotient, a Smith number, an admirable number, and the sum of four consecutive primes (181 + 191 + 193 + 197).

There are 762 1's in all partitions of 25 into odd parts There are Six nines in the decimal representation of pi after the 762nd digit.

763 = 7 × 109.

It is the sum of nine consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103). There are 763 degree-8 permutations of order exactly 2.

764 = 22 × 191. It is a telephone number.

765=32 × 5 × 17.

It is an octagonal pyramidal number.

It is a Japanese word-play for Namco.

766 = 2 × 383. It is a centered pentagonal number, a nontotient, and the sum of twelve consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89).

767 = 13 × 59. It is a palindromic number and a Thabit number because 767 = 28 × 3 − 1

768 = 28 × 3. It is the sum of eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109).

769 is a prime number, a Chen prime, a lucky prime, and a Proth prime.

770s

770 = 2 × 5 × 7 × 11. It is a nontotient and a Harshad number.

∑

n

=

0

10

770

n

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

is prime

It holds special importance in the Chabad-Lubavitch Hasidic movement.

771 = 3 × 257.

It is 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 is a prime number, an Eisenstein prime with no imaginary part, a prime index prime, and a tetranacci number.

780s

780 = 22 × 3 × 5 × 13. It is a hexagonal number, a Harshad number, and the 39th triangular number. It is the sum of four consecutive primes in a quadruplet (191, 193, 197, and 199) and the sum of ten consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101).

780 and 990 are the fourth smallest pair of triangular numbers whose sum and difference (1770 and 210) are also triangular.

781 = 11 × 71. It is a zero of the Mertens function and a lazy caterer number (sequence A000124 in the OEIS) It is the sum of powers of 5, or equivalently, repdigit in base 5 (11111)

782 = 2 × 17 × 23. It is a sphenic number, a nontotient, a pentagonal number, and a Harshad number.

783 = 33 × 29. It is a heptagonal number.

784 = 24 × 72. It is a happy number.

Since 784=282, 784 is a perfect square. It is the sum of the cubes of the first seven positive integers;

1

3

+

2

3

+

3

3

+

4

3

+

5

3

+

6

790s

790 = 2 × 5 × 79. It is a sphenic number, a nontotient, an aspiring number, and the aliquot sum of 1574. It is a Harshad number in bases 2, 7, 14, and 16.

791 = 7 × 113. It is a centered tetrahedral number, the sum of the first twenty-two primes, and the sum of seven consecutive primes (101 + 103 + 107 + 109 + 113 + 127 + 131).

792 = 23 × 32 × 11. It is a Harshad number.

the sum of the nontriangular numbers between successive triangular numbers

There are 792 integer partitions of 21.

792=

(

12

5

)

{\displaystyle {\tbinom {12}{5}}}

, a binomial coefficient.

793 = 13 × 61. It is a zero of the Mertens function, a star number, and a happy number.

794 = 2 × 397. It is a nontotient.

794= 16 + 26 + 36.

795 = 3 × 5 × 53. It is a sphenic number and a zero of the Mertens function.

There are 795 permutations of length 7 with 2 consecutive ascending pairs.

796 = 22 × 199. It is a zero of the Mertens function and the sum of six consecutive primes (113 + 127 + 131 + 137 + 139 + 149).