# 91 (number)

 ← 90 91 92 →
Cardinal ninety-one
Ordinal 91st
(ninety-first)
Factorization 7 × 13
Divisors 1, 7, 13, 91
Greek numeral ϞΑ´
Roman numeral XCI
Binary 10110112
Ternary 101013
Quaternary 11234
Quinary 3315
Senary 2316
Octal 1338
Duodecimal 7712
Vigesimal 4B20
Base 36 2J36

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

## In mathematics

91 is:

• the twenty-seventh distinct semiprime and the second of the form (7.q).
• a triangular number.
• a hexagonal number,[1] one of the few such numbers to also be a centered hexagonal number.[2]
• a centered nonagonal number.[3]
• a centered cube number.[4]
• a square pyramidal number, being the sum of the squares of the first six integers.[5]
• 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.[6]
• a repdigit in base 9 (1119).
• palindromic in bases 3 (101013), 9 (99), and 12 (7712).
• the fourth composite number in the 11-aliquot tree. The aliquot sum of 91 is 21 within the aliquot sequence (91,21,11,1,0).
• 91 (of 365) days of the year have a number divisible by 3.

The decimal equivalent of the fraction ​191 can be obtained by using powers of 9.

## In other fields

Ninety-one is also:

## References

1. ^ "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
2. ^ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
3. ^ "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.
4. ^ "Sloane's A005898 : Centered cube numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
5. ^ "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.