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<ref>Template:Cite OEIS</ref> 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.<ref>Template:Cite web</ref>
• a hexagonal number,<ref>Template:Cite web</ref> one of the few such numbers to also be a centered hexagonal number.<ref>Template:Cite web</ref>
• a centered nonagonal number.<ref>Template:Cite web</ref>
• a centered cube number.<ref>Template:Cite web</ref>
• a square pyramidal number, being the sum of the squares of the first six integers.<ref>Template:Cite web</ref>
• 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):<ref>Template:Cite OEIS</ref>
  Template:Nowrap. (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:
  Template:Nowrap.
• The only other ways to write 91 as a sum of distinct squares are:
  Template:Nowrap and Template:Nowrap.
• the smallest pseudoprime satisfying the congruence Template:Nowrap.<ref>Friedman, Erich. What's Special About This Number? Template:Webarchive</ref>
• a repdigit in base 9 (111₉).
• palindromic in bases 3 (10101₃), 9 (111₉), and 12 (77₁₂).
• a Riordan number.<ref>Template:Cite web</ref>
• the smallest number that looks prime but is not, proven using the Rotten Theorem by John Conway.<ref>Template:Cite web</ref><ref>Template:Cite web</ref>

The decimal equivalent of the fraction Template:Frac can be obtained by using powers of 9.

In science

• McCarthy 91 function, a recursive function in discrete mathematics

References

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