91 (ninety-one) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 90 and preceding 92.

In mathematics

91 is: * the twenty-seventh distinct semiprime and the second of the form (7.q), where q is a higher prime. * the

aliquot sum In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. That is, s(n)=\sum_ d \, . It can be used to characterize the prime numbers, perfect numbers, sociabl ...

of 91 is 21; itself a semiprime, within an

aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Def ...

of two composite numbers (91, 21, 11, 1, 0) to the prime in the 11-aliquot tree. 91 is the fourth composite number in the 11-aliquot tree. (91, 51, 21, 18). * the 13th

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

. * a

hexagonal number A hexagonal number is a figurate number. The ''n''th hexagonal number ''h'n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so ...

, one of the few such numbers to also be a

centered hexagonal number In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, is a centered polygonal number, centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot ...

. * a centered nonagonal number. * a centered cube number. * a square pyramidal number, being the sum of the squares of the first six

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s. * the smallest positive integer expressible as a

sum of two cubes In mathematics, the sum of two cubes is a cubed number added to another cubed number. Factorization Every sum of cubes may be factored according to the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2) in elementary algebra. Binomial numbers g ...

in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes):
. (See

1729 Events January–March * January 8 – Frederick, the eldest son of King George II of Great Britain is made Prince of Wales at the age of 21, a few months after he comes to Britain for the first time after growing up in Hanover ...

for more details).
This implies that 91 is the second cabtaxi number. * the smallest positive integer expressible as a sum of six distinct squares:
. * The only other ways to write 91 as a sum of distinct squares are:
and * . * the smallest pseudoprime satisfying the congruence . * a

repdigit In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Ex ...

in base 9 (111₉). * palindromic in bases 3 (10101₃), 9 (111₉), and 12 (77₁₂). * a Riordan number. * the smallest number that looks

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

but is not, proven using the Rotten Theorem by John Conway. The decimal equivalent of the fraction can be obtained by using powers of 9.

In science

* McCarthy 91 function, a recursive function in

discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

References

{{DEFAULTSORT:91 (Number) Integers