800 (number)

801

801 is a Harshad number. 801 is the sum of a square and positive cube in more than one way, and a sum of distinct positive cubes in more than one way:^[2][3]

${\displaystyle 801=17^{2}+8^{3}=26^{2}+5^{3}=1^{3}+2^{3}+4^{3}+6^{3}+8^{3}=2^{3}+4^{3}+9^{3}.}$

In the gematria of 2nd century bishop Irenaeus, 801 stands for both the Greek word for a dove, and for Alpha and Omega, and therefore represents God in two ways.^[4]

There are 801 club patterns in a 50x50 grid of coins.^[5]

802

802 = 2 × 401. It is a nontotient, a happy number, the sum of eight consecutive primes (83 + 89 + 97 + 101 + 103 + 107 + 109 + 113), and the sum of 4 consecutive triangular numbers^[6] (171 + 190 + 210 + 231).

803

803 = 11 × 73. It is a Harshad number, the sum of three consecutive primes (263 + 269 + 271), and the sum of nine consecutive primes (71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107).

There are 803 partitions of 34 into Fibonacci parts.^[7]

804

804 = 2² × 3 × 67. It is a nontotient, a Harshad number, and a refactorable number.^[8]

"The 804" is a local nickname for the Greater Richmond Region of the U.S. state of Virginia, derived from its telephone area code (although the area code covers a larger area).^[9][10]

805

805 = 5 × 7 × 23. It is a sphenic number. There are 805 partitions of 38 into nonprime parts^[11]

806

806 = 2 × 13 × 31. It is a sphenic number, a nontotient, a happy number, and the totient sum for first 51 integers.

806 = Phi(51)^[12]

807

807 = 3 × 269 = antisigma(42)^[13]

808

808 = 2³ × 101. It is a refactorable number and a strobogrammatic number.^[14]

809

809 is a prime number, a Sophie Germain prime,^[15] a Chen prime, and an Eisenstein prime with no imaginary part.

810

810 = 2 × 3⁴ × 5. It is a harshad number. There are 810 non-equivalent ways of expressing 100,000 as the sum of two prime numbers.^[16] Theree are 810 distinct reduced words of length 5 in the Coxeter group of "Apollonian reflections" in three dimensions.^[17]

811

811 is a prime number, a twin prime, a Chen prime, the largest minimal prime in base 9, and the sum of five consecutive primes (151 + 157 + 163 + 167 + 173). It is a happy number and a zero of Mertens function.

812

812 = 2² × 7 × 29. It is an admirable number, a pronic number,^[18] a balanced number,^[19] and a zero of Mertens function.

813

813 = 3 × 271. It is a Blum integer.^[20]

814

814 = 2 × 11 × 37. It is a sphenic number, a nontotient, and a zero of Mertens function. There are 814 fixed hexahexes.

815

815 = 5 × 163. There are 815 graphs with 8 vertices and a distinguished bipartite block.^[21]

816

816 = 2⁴ × 3 × 17. It is a tetrahedral number,^[22] a Padovan number,^[23] and a Zuckerman number.

817

817 = 19 × 43. It is a centered hexagonal number^[24] and the sum of three consecutive primes (269 + 271 + 277).

818

818 = 2 × 409. It is a nontotient and a strobogrammatic number^[14]

819

819 = 3² × 7 × 13. It is a square pyramidal number.^[25]

820

820 = 2² × 5 × 41. It is a Harshad number, a happy number, a repdigit (1111) in base 9, and the 40th triangular number.^[26]

821

821 is a prime number, a twin prime, a Chen prime, and an Eisenstein prime with no imaginary part. It forms a prime quadruplet with 823, 827, and 829. It is a lazy caterer number.^[27]

822

822 = 2 × 3 × 137. It is a sphenic number, a member of the Mian–Chowla sequence,^[28] and the sum of twelve consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97).

823

823 is a prime number, a twin prime, and a lucky prime. It forms a prime quadruplet with 821, 827, and 829. It is a zero of Mertens function.

824

824 = 2³ × 103, refactorable number, nontotient, a zero of Mertens function, and the sum of ten consecutive primes (61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103).

825

825 = 3 × 5² × 11. It is a Smith number,^[29] a Harshad number, and a zero of Mertens function.

826

826 = 2 × 7 × 59. It is a sphenic number. There are 825 partitions of 29 into parts each of which is used a different number of times.^[30]

827

827 is a prime number, a twin prime, a Chen prime, a Eisenstein prime with no imaginary part, and the sum of seven consecutive primes (103 + 107 + 109 + 113 + 127 + 131 + 137). It forms a prime quadruplet with 821, 823, and 829. It is a strictly non-palindromic number.^[31]

828

828 = 2² × 3² × 23. It is a Harshad number and a triangular matchstick number.^[32]

829

829 is a prime number, a twin prime, a Chen prime, and the sum of three consecutive primes (271 + 277 + 281). It forms a prime quadruplet with 821, 823, and 827. It is a centered triangular number.

830

830 = 2 × 5 × 83. It is a sphenic number, a nontotient, the totient sum of the first 52 integers, and the sum of four consecutive primes (197 + 199 + 211 + 223).

831

831 = 3 × 277. There are 831 partitions of 32 into at most 5 parts.^[33]

832

832 = 2⁶ × 13. It is a Harshad number and a member of the Horadam sequence (0,1,4,2).^[34]

833

833 = 7² × 17. It is an octagonal number^[35] and a centered octahedral number.^[36]

834

834 = 2 × 3 × 139. It is a cake number, a sphenic number, a nontotient, and the sum of six consecutive primes (127 + 131 + 137 + 139 + 149 + 151).

835

835 = 5 × 167. It is a Motzkin number.^[37]

836

The factorization of 836 is 2² × 11 × 19, so its proper factors are 1, 2, 4, 11, 19, 22, 38, 44, 76, 209, and 418. They sum to 844. As this is greater than 836, it is an abundant number, but no subset sums to 836, so it is not a semiperfect number; therefore it is a weird number.^[38] Besides, 836 is the smallest weird number that is also an untouchable number, i.e. there is no n such that the sum of proper factors of n equals 836. (The only smaller weird number, 70, is not untouchable, since σ(134) − 134 = 70).

837

837 = 3³ × 31. It is the 36th generalized heptagonal number.^[39]

838

838 = 2 × 419. It is a palindromic number. There are 838 distinct products ijk with 1 <= i<j<k <= 23.^[40]

839

839 is a prime number, a safe prime,^[41] a Chen prime, an Eisenstein prime with no imaginary part, and the sum of five consecutive primes (157 + 163 + 167 + 173 + 179). It is a highly cototient number.^[42]

840

841

841 = 29² = 20² + 21², sum of three consecutive primes (277 + 281 + 283), sum of nine consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109), centered square number,^[43] centered heptagonal number,^[44] centered octagonal number^[45]

842

842 = 2 × 421. It is a nontotient. There are 842 series-reduced trees with 18 nodes.^[46]

842!! - 1 is prime.^[47]

843

843 = 3 × 281. It is a Lucas number.^[48]

844

844 = 2² × 211. It is a nontotient. It is the smallest 5 consecutive integers which are not squarefree:

844 = 2² × 211, 845 = 5 × 13², 846 = 2 × 3² × 47, 847 = 7 × 11², and 848 = 2⁴ × 53.^[49]

845

845 = 5 × 13². It is a concentric pentagonal number.^[50]There are 845 emergent parts in all partitions of 22.^[51]

846

846 = 2 × 3² × 47. It is a nontotient, a Harshad number, and the sum of eight consecutive primes (89 + 97 + 101 + 103 + 107 + 109 + 113 + 127).

847

847 = 7 × 11². It is a happy number. There are 847 partitions of 29 that do not contain 1 as a part.^[52]

848

848 = 2⁴ × 53. It is an untouchable number.

849

849 = 3 × 283, It is a zero of Mertens function and a Blum integer.

850

850 = 2 × 5² × 17. It is a zero of Mertens function and a nontotient. The sum of the squares of the divisors of 26 is 850 (sequence A001157 in the OEIS).

The maximum possible Fair Isaac credit score is 850.

851

851 = 23 × 37 There are 851 compositions of 18 into distinct parts^[53]

852

852 = 2² × 3 × 71. It is a pentagonal number^[54] and a Smith number.^[29]

853

853 is a prime number, a Perrin number,^[55] a zero of Mertens function, and a strictly non-palindromic number. The average of first 853 prime numbers is an integer (sequence A045345 in the OEIS). There are 853 connected graphs with 7 nodes.

854

854 = 2 × 7 × 61. It is a sphenic number and a nontotient. There are 854 unlabeled planar trees with 11 nodes.^[56]

855

855 = 3² × 5 × 19. It is a decagonal number^[57] and a centered cube number.^[58]

856

856 = 2³ × 107. It is a nonagonal number,^[59] a centered pentagonal number,^[60] and a refactorable number.

857

857 is a prime number, a Chen prime, an Eisenstein prime with no imaginary part, and the sum of three consecutive primes (281 + 283 + 293).

858

858 = 2 × 3 × 11 × 13. It is a Giuga number.^[61]

859

859 is a prime number and a prime index prime. There are 859 planar partitions of 11.^[62]

860

860 = 2² × 5 × 43. It is aHoax number^[63] and the sum of four consecutive primes (199 + 211 + 223 + 227).

861

861 = 3 × 7 × 41. It is a sphenic number, a hexagonal number,^[64] a Smith number,^[29] and the 41st triangular number.^[26]

862

862 = 2 × 431. It is a lazy caterer number.^[65]

863

863 is a prime number, a safe prime,^[41] a Chen prime, an Eisenstein prime with no imaginary part, and an index of a prime Lucas number.^[66]It is the sum of five consecutive primes (163 + 167 + 173 + 179 + 181) and the sum of seven consecutive primes (107 + 109 + 113 + 127 + 131 + 137 + 139).

864

864 = 2⁵ × 3³. It is an Achilles number and a Harshad number. It the sum of a twin prime pair (431 + 433) and the sum of six consecutive primes (131 + 137 + 139 + 149 + 151 + 157).

865

865 = 5 × 173.

866

866 = 2 × 433. It is a nontotient and the number of cubes of edge length 1 required to make a hollow cube of edge length 13. There are 866 one-sided noniamonds.^[67]

867

867 = 3 × 17². There are 867 5-chromatic simple graphs on 8 nodes.^[68]

868

868 = 2² × 7 × 31 = J₃(10).^[69] It is a nontotient.

869

869 = 11 × 79. It is a zero of Mertens function.

870

870 = 2 × 3 × 5 × 29. It is a pronic number,^[18] a nontotient, a sparsely totient number,^[70] a Harshad number, and the sum of ten consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107). It is the magic constant of n×n normal magic square and n-queens problem for n = 12.

871

871 = 13 × 67. It is the thirteenth tridecagonal number.

872

872 = 2³ × 109. It is a refactorable number and anontotient.

872! + 1 is prime.

873

873 = 3² × 97 = 1! + 2! + 3! + 4! + 5! + 6!.

874

874 = 2 × 19 × 23 = 0! + 1! + 2! + 3! + 4! + 5! + 6!. It is a sphenic number, a nontotient, a Harshad number, a happy number, and the sum of the first twenty-three primes.

875

875 = 5³ × 7. It can be uniquely expressed as a difference of two positive cubes: 875 = 10³ – 5³.^[71]

876

876 = 2² × 3 × 73. It is a generalized pentagonal number.^[72]

877

877 is a prime number, a prime index prime, a Chen prime, a zero of Mertens function, a Bell number,^[73] and a strictly non-palindromic number.^[31]

878

878 = 2 × 439. It is a nontotient. There are 878 Pythagorean triples with a hypotenuse less than 1000.^[74]

879

879 = 3 × 293. It is a candidate Lychrel seed number. There are 879 regular hypergraphs spanning 4 vertices.^[75]

880

880 = 2⁴ × 5 × 11 = 11!!!.^[76] It is a Harshad number and a 148-gonal number. There are 880 n×n magic squares for n = 4.^[77]

881

881 is a prime number, a twin prime, a Chen prime, an Eisenstein prime with no imaginary part, and the sum of nine consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113). It is a happy number.

881 is a bilingual play on words when text chatting in Mandarin Chinese or bilingually Mandarin Chinese and English. "881" is pronounced ba ba yi in Mandarin, and thus puns on "bye-bye." Probably an elaboration of the similar pun on "88" (ba-ba). See 88 (number).^{[citation needed]}

882

882 = 2 × 3² × 7² = ${\displaystyle {\binom {9}{5}}_{2}}$ It is a trinomial coefficient,^[78] a Harshad number, and the totient sum of the first 53 integers.

883

883 is a prime number, a twin prime, a lucky prime, the sum of three consecutive primes (283 + 293 + 307), and the sum of eleven consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103). It is a zero of Mertens function.

884

884 = 2² × 13 × 17. It is a zero of Mertens function. There are 884 points on surface of tetrahedron with sidelength 21.^[79]

885

885 = 3 × 5 × 59. It is a sphenic number. There are 885 series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of 7.^[80]

886

886 = 2 × 443. It is a zero of Mertens function.

887

887 is a prime number followed by primal gap of 20, a safe prime,^[41] a Chen prime, and an Eisenstein prime with no imaginary part. It is the first iteration of the 196 trajectory (196 + 691 = 887).

888

889

889 = 7 × 127. It is a zero of Mertens function.

890

890 = 2 × 5 × 89. It is a sphenic number, a nontotient, and the sum of four consecutive primes (211 + 223 + 227 + 229). 890 is the sum of the squares of two successive primes: 890 = 19² + 23².^[81]

891

891 = 3⁴ × 11. It is an octahedral number and the sum of five consecutive primes (167 + 173 + 179 + 181 + 191).

892

892 = 2² × 223. It is a nontotient. There are 892 regions formed by drawing the line segments connecting any two perimeter points of a 6 times 2 grid of squares like this (sequence A331452 in the OEIS).

893

893 = 19 × 47. It is a zero of Mertens function.

893 is considered an unlucky number in Japan, because its digits read sequentially are the literal translation of yakuza.

894

894 = 2 × 3 × 149. It is a sphenic number and a nontotient.

895

895 = 5 × 179. It is a Smith number,^[29] a Woodall number,^[82] and a zero of Mertens function.

896

896 = 2⁷ × 7. It is a refactorable number, a zero of Mertens function, and the sum of six consecutive primes (137 + 139 + 149 + 151 + 157 + 163).

897

897 = 3 × 13 × 23. It is a sphenic number and a Cullen number (sequence A002064 in the OEIS).

898

898 = 2 × 449. It is a nontotient and a zero of Mertens function.

899

899 = 29 × 31. It is the product of a pair of twin primes,^[83] a happy number, and the smallest number with digit sum 26.^[84] There are 899 partitions of 51 into prime parts.