91

Number
$91$ (ninety-one) is:


 * $7 \times 13$


 * The $30$th semiprime:
 * $91 = 7 \times 13$


 * The $13$th triangular number after $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78$:
 * $91 = \displaystyle \sum_{k \mathop = 1}^{13} k = \dfrac {13 \times \left({13 + 1}\right)} 2$


 * The $7$th hexagonal number after $1, 6, 15, 28, 45, 66$:
 * $91 = 1 + 5 + 9 + 13 + 17 + 21 + 25 = 7 \left({2 \times 7 - 1}\right)$


 * The $6$th centered hexagonal number after $1, 7, 19, 37, 61$:
 * $91 = 1 + 6 + 12 + 18 + 24 + 30 = 6^3 - 5^3$


 * The $6$th square pyramidal number after $1$, $5$, $14$, $30$, $55$:
 * $91 = 1 + 4 + 9 + 16 + 25 + 36 = \dfrac {6 \left({6 + 1}\right) \left({2 \times 6 + 1}\right)} 6$


 * The $3$rd after $1, 55$ of the $4$ square pyramidal numbers which are also triangular.


 * The $17$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$:
 * $91 \to 9^2 + 1^2 = 81 + 1 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$


 * The $1$st Fermat pseudoprime to base $3$:
 * $3^{91} \equiv 3 \pmod {91}$


 * The $17$th positive integer $n$ after $5, 11, 17, 23, 29, 30, 36, 42, 48, 54, 60, 61, 67, 73, 79, 85$ such that no factorial of an integer can end with $n$ zeroes.


 * The $8$th positive integer which cannot be expressed as the sum of a square and a prime:
 * $1, 10, 25, 34, 58, 64, 85, 91, \ldots$


 * The $5$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
 * $10, 34, 58, 85, 91, \ldots$


 * The $2$nd of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
 * $73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$

Also see

 * 91 is Smallest Fermat Pseudoprime Base 3