91 (number)

Template:Infobox number 91 (ninety-one) is the natural number following 90 and preceding 92.

In mathematics

File:Square-pyramidal-91.png

91 is a square pyramidal number.

File:Centered-hexagonal-91.png

91 is a centered hexagonal number.

File:Cube-sum-diff-91.png

91 as the sum and difference of two positive cubes

91 is:

• the twenty-seventh distinct semiprime and the second of the form (7.q), where q is a higher prime.
• the aliquot sum of 91 is 21; itself a semiprime, within an aliquot sequence 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 hexagonal number, one of the few such numbers to also be a centered hexagonal number.
• a centered nonagonal number.
• a centered cube number.
• a square pyramidal number, being the sum of the squares of the first six integers.
• 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).
  This implies that 91 is the second cabtaxi number.
• 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.
• the smallest pseudoprime satisfying the congruence 3ⁿ ≡ 3 mod n.
• a repdigit in base 9 (111₉).
• palindromic in bases 3 (10101₃), 9 (111₉), and 12 (77₁₂).
• a Riordan number.
• the smallest number that looks prime but is not, proven using the Rotten Theorem by John Conway.

The decimal equivalent of the fraction 1⁄91 can be obtained by using powers of 9.

In science

• McCarthy 91 function, a recursive function in discrete mathematics