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91 (number)
Natural number

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

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In mathematics

91 is:

  • the twenty-seventh distinct semiprime1 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.2
  • a hexagonal number,3 one of the few such numbers to also be a centered hexagonal number.4
  • a centered nonagonal number.5
  • a centered cube number.6
  • a square pyramidal number, being the sum of the squares of the first six integers.7
  • 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):8 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.9
  • a repdigit in base 9 (1119).
  • palindromic in bases 3 (101013), 9 (1119), and 12 (7712).
  • a Riordan number.10
  • the smallest number that looks prime but is not, proven using the Rotten Theorem by John Conway.1112

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

In science

References

  1. Sloane, N. J. A. (ed.). "Sequence A001358". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩

  2. "A000217 - OEIS". oeis.org. Retrieved 2024-11-28. https://oeis.org/A000217 ↩

  3. "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29. https://oeis.org/A000384 ↩

  4. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29. https://oeis.org/A003215 ↩

  5. "Sloane's A060544 : Centered 9-gonal (also known as nonagonal or enneagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29. https://oeis.org/A060544 ↩

  6. "Sloane's A005898 : Centered cube numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29. https://oeis.org/A005898 ↩

  7. "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29. https://oeis.org/A000330 ↩

  8. Sloane, N. J. A. (ed.). "Sequence A047696 (Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩

  9. Friedman, Erich. What's Special About This Number? Archived 2018-02-23 at the Wayback Machine https://www.stetson.edu/~efriedma/numbers.html ↩

  10. "Sloane's A005043 : Riordan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29. https://oeis.org/A005043 ↩

  11. "John Conway proves that 91 is the smallest number which looks prime but isn't". Ryan Andersen. 31 December 2020. Retrieved 2024-05-09. https://www.youtube.com/watch?v=S75VTAGKQpk ↩

  12. "Prime Numbers". 9 May 2024. https://aperiodical.com/tag/prime-numbers/ ↩