91 (number)

← 90 91 92 →
← 90 91 92 93 94 95 96 97 98 99 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	ninety-one
Ordinal	91st (ninety-first)
Factorization	7 × 13
Divisors	1, 7, 13, 91
Greek numeral	ϞΑ´
Roman numeral	XCI, xci
Binary	10110112
Ternary	101013
Senary	2316
Octal	1338
Duodecimal	7712
Hexadecimal	5B16

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

In mathematics

91 is a square pyramidal number.

91 is a centered hexagonal number.

91 as the sum and difference of two positive cubes

91 is:

• the twenty-seventh distinct semiprime ^[1] 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. ^{[11] [12]}

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

References

1. Sloane, N. J. A. (ed.). "SequenceA001358". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. "A000217 - OEIS". oeis.org. Retrieved 2024-11-28.
3. "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
4. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
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.
6. "Sloane's A005898 : Centered cube numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
7. "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
8. Sloane, N. J. A. (ed.). "SequenceA047696(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.
9. Friedman, Erich. What's Special About This Number? Archived 2018-02-23 at the Wayback Machine
10. "Sloane's A005043 : Riordan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
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.
12. "Prime Numbers". 9 May 2024.