700 (number)

701

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

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

703

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

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

705

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

706

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

707

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

708 = 2² × 3 × 59. There are 708 partitions of 28 that do not contain 1 as a part.

709

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.

710

710 = 2 × 5 × 71. It is a sphenic number and a nontotient. There are 710 forests with 11 vertices.

711

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

712

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

713 = 23 × 31. It is a Blum integer.

In Judaism there are 713 letters on a Mezuzah scroll.

714

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

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=${\displaystyle {\tbinom {13}{4}}}$.

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

716

716 = 2² × 179.

717

717 = 3 × 239. It is a palindromic number.

718

718 = 2 × 359.

719

719 is a prime number, a Sophie Germain prime, a safe prime, a Chen prime, and an Eisenstein prime with no imaginary part.

Because 719 = 6! − 1, 719 is a factorial prime.

It is the sum of seven consecutive primes (89 + 97 + 101 + 103 + 107 + 109 + 113).

720

721

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

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

723

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

724

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

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

726

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

727

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

728

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

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.

730

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

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

732

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

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

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

735

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

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

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

738

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

739

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

740

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

741

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

742

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.

743

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

744 is 2*2*2*3*3, 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: $${\displaystyle j(\tau )=q^{-1}+744+196\,884q+21\,493\,760q^{2}+864\,299\,970q^{3}+\cdots ,}$$where ${\displaystyle q=e^{2\pi i\tau }}$. One consequence of this is that 744 appears in expressions for Ramanujan's constant and other almost integers.

745

745 = 5 × 149. There are 745 non-connected simple labeled graphs covering 6 vertices.

746

746 = 2 × 373.

It is a nontotient. There are 746 non-normal semi-magic squares with sum of entries equal to 6.

746=1⁵ + 2⁴ + 3⁶ = 1⁷ + 2⁴ + 3⁶.

747

747 = 3² × 83. It is a palindromic number.

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

749

749 = 7 × 107. It is a blum integer and the sum of three consecutive primes (241 + 251 + 257).

750

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

751

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

752

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

753

753 = 3 × 251. It is a blum integer.

754

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

755 = 5 × 151. There are 755 vertices in a regular drawing of the complete bipartite graph K_9,9.

756

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

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

758 = 2 × 379.

It is a nontotient and a prime number of measurement.

759

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).

760

760 = 2³ × 5 × 19.

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

761

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

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

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

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

765

765=3² × 5 × 17.

It is an octagonal pyramidal number.

It is a Japanese word-play for Namco.

766

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

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

768

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

769

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

770

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

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

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

771

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 ${\displaystyle \cos \left({\frac {2\pi }{771}}\right)}$ can be written in terms of square roots.

772

772 = 2² × 193.

772!!!!!!+1 is prime.

773

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

• the sum of the number of cells that make up the convex, regular 4-polytopes

774

774 = 2 × 3² × 43. It is a nontotient, a Harshad number, and the totient sum for first 50 integers

775

775 = 5² × 31. It is a member of the Mian–Chowla sequence

776

776 = 2³ × 97.

It is a refactorable number. There are 776 compositions of 6 whose parts equal to q can be of q² kinds.

777

778

778 = 2 × 389. It is a nontotient and a Smith number.

779

779 = 19 × 41. It is a highly cototient number.

780

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

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

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

783

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

784

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; ${\displaystyle 1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}+7^{3}=784}$.

785

785 = 5 × 157. It is a zero of the Mertens function. There are 785 series-reduced planted trees with 6 leaves of 2 colors.

786

787

787 is a prime number, a Chen prime, a lucky prime, a palindromic prime, and the sum of five consecutive primes (149 + 151 + 157 + 163 + 167).

788

788 = 2² × 197.

It is a nontotient. There are 788 compositions of 12 into parts with distinct multiplicities.

789

789 = 3 × 263. It is a Blum integer and the sum of three consecutive primes (257 + 263 + 269).

790

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

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

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=${\displaystyle {\tbinom {12}{5}}}$, a binomial coefficient.

793

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

794

794 = 2 × 397. It is a nontotient.

794= 1⁶ + 2⁶ + 3⁶.

795

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

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

797

797 is a prime number, a Chen prime, an Eisenstein prime with no imaginary part, a palindromic prime, a two-sided prime, and a prime index prime.

798

798 = 2 × 3 × 7 × 19. It is a zero of the Mertens function and a nontotient.

• the product of primes indexed by the prime exponents of 10!

799

799 = 17 × 47. It is the smallest number with digit sum 25