75

Number
$75$ (seventy-five) is:


 * $3 \times 5^2$


 * The $5$th pentagonal pyramidal number after $1, 6, 12, 40$:
 * $75 = 1 + 5 + 12 + 22 + 35 = \dfrac {5^2 \left({5 + 1}\right)} 2$


 * The $10$th trimorphic number after $1, 4, 5, 6, 9, 24, 25, 49, 51$:
 * $75^3 = 421 \, 8 \mathbf {75}$


 * The index (after $2, 3, 6, 30$) of the $5$th Woodall prime:
 * $75 \times 2^{75} - 1$


 * The $18$th lucky number:
 * $1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 73, 75, \ldots$


 * The $6$th Keith number after $14, 19, 28, 47, 61$:
 * $7, 5, 12, 17, 29, 46, 75, \ldots$


 * The $1$st of the $2$nd pair of consecutive integers which both have $6$ divisors:
 * $\tau \left({75}\right) = \tau \left({76}\right) = 6$


 * With $48$, an element of the $1$st quasiamicable pair:
 * $\sigma \left({48}\right) = \sigma \left({75}\right) = 124 = 48 + 75 + 1$


 * The $3$rd of the $2$nd ordered quadruple of consecutive integers that have sigma values which are strictly increasing:
 * $\sigma \left({73}\right) = 74$, $\sigma \left({74}\right) = 114$, $\sigma \left({75}\right) = 124$, $\sigma \left({76}\right) = 140$


 * The $4$th inconsummate number after $62, 63, 65$:
 * $\nexists n \in \Z_{>0}: n = 75 \times s_{10} \left({n}\right)$


 * The $6$th positive integer $n$ after $4, 7, 15, 21, 45$ such that $n - 2^k$ is prime for all $k$