GOTW- Counting - Page 12

Ninety-one is the twenty-seventh distinct semiprime and the second of the form (7.q). The aliquot sum of 91 is 21 within the aliquot sequence (91,21,11,1,0) 91 being the fourth composite number in the 11-aliquot tree.

Ninety-one is a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also acentered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers.

It is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6³+(-5)³ = 4³+3³. (See 1729 for more details).

It is also the smallest positive integer expressible as a sum of six distinct squares: 91 = 1²+2²+3²+4²+5²+6². The only other ways to write 91 as a sum of distinct squares are: 91 = 1²+4²+5²+7² and 91 = 1²+3²+9².

It is also the smallest pseudoprime satisfying the congruence 3ⁿ = 3 mod n.^[1]

91 is a repdigit in base 9 (111).

Ninety-two is a pentagonal number.

There are 92 Johnson solids. The snub dodecahedron has 92 faces, the most an Archimedean solid can have.

For n = 8, there are 92 solutions in the n-Queens Problem.

Since it is possible to find sequences of 92 consecutive integers such that each inner member member shares a factor with either the first or the last member, 92 is an Erdos–Woods number.

There are 92 "atomic elements" in the Look-and-say sequence, corresponding to the 92 non-transuranic elements in the chemist's periodic table.

Ninety-three is the twenty-eighth distinct semiprime and the ninth of the form (3.q). The aliquot sum of 93 is 35 within the aliquot sequence (93,35,13,1,0)93 being the fourth composite number in the 13-aliquot tree. 93 is the first number in the third triplet of three consecutive distinct semiprimes, 93, 94 and 95.

Since its two prime factors, 3 and 31 are both Gaussian primes, this means that 93 is a Blum integer.

Given 93 the Mertens function returns 0.

There are 93 solutions to Znm's problem for length 8.

93 is a repdigit in base 5 (333).

Ninety-four is the twenty-ninth distinct semiprime and the fourteenth of the form (2.q). The aliquot sum of 94 is 50 within the aliquot sequence (94,50,43,1,0) 94 being the ninth composite number in the 43-aliquot tree. 94 is the second number in the third triplet of three consecutive distinct semiprimes, 93, 94 and 95.

94 is a 17-gonal number and a nontotient.

Since it is possible to find sequences of 94 consecutive integers such that each inner member member shares a factor with either the first or the last member, 94 is an Erdos–Woods number.

In decimal, 94 is a Smith number.

Lik 93 edit it.. You have posted 91 twice 😆

Ninety-six is an octagonal number, a refactorable number and an untouchable number. Since it is a multiple of 6, it is a semiperfect number

Ninety-six is the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126, the previous being24.

The sum of Euler's totient function f(x) over the first seventeen integers is 96.

Since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with either the first or the last member, 96 is an Erdos–Woods number.

Every integer greater than 96 may be represented as a sum of distinct super-prime numbers.

97 is the 25th prime number (the largest two-digit prime number in base 10), following 89 and preceding 101. 97 is a Proth prime as it is 3 2⁵ + 1.

The numbers 97, 907, 9007, 90007 and 900007 are happy primes. However, 9000007 (read as nine million seven) is composite and has the factorisation277 32491.

97 is the tenth member of the Mian–Chowla sequence.

97 is the smallest factor of one more than the product of the first twenty-five terms of the Euclid–Mullin sequence, making it the twenty-sixth term.

Since there is no integer that added to its own digits adds up to 97, 97 is a self number in base 10.