# 75

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## Number

$75$ (seventy-five) is:

$3 \times 5^2$

With $48$, an element of the $1$st quasiamicable pair:
$\map {\sigma_1} {48} = \map {\sigma_1} {75} = 124 = 48 + 75 + 1$

The $1$st of the $2$nd pair of consecutive integers which both have $6$ divisors:
$\map {\sigma_0} {75} = \map {\sigma_0} {76} = 6$

The $3$rd of the $2$nd ordered quadruple of consecutive integers that have divisor sums which are strictly increasing:
$\map {\sigma_1} {73} = 74$, $\map {\sigma_1} {74} = 114$, $\map {\sigma_1} {75} = 124$, $\map {\sigma_1} {76} = 140$

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

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

The $5$th tri-automorphic number after $2$, $5$, $7$, $67$:
$75^2 \times 3 = 16 \, 8 \mathbf {75}$

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

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

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

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

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 $29$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $\ldots$

### Arithmetic Functions on $75$

 $\ds \map {\sigma_0} { 75 }$ $=$ $\ds 6$ $\sigma_0$ of $75$ $\ds \map {\sigma_1} { 75 }$ $=$ $\ds 124$ $\sigma_1$ of $75$