# 81

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## Number

$81$ (eighty-one) is:

$3^4$

The $1$st square number which is also heptagonal:
$81 = \dfrac {6 \paren {5 \times 6 - 3} } 2 = 9^2$

The $2$nd power of $9$ after $(1)$, $9$:
$81 = 9^2$

The $2$nd, after $1$, of the $4$ harshad numbers which can each be expressed as the product of the sum of its digits and the reversal of the sum of its digits:
$81 = 9 \times 9 = 9 \times \paren {1 + 8}$

The $3$rd and last number after $0$, $1$ whose square root equals the sum of its digits:
$\sqrt {81} = 9 = 8 + 1$

The $3$rd fourth power after $1$, $16$:
$81 = 3 \times 3 \times 3 \times 3$

The $4$th power of $3$ after $(1)$, $3$, $9$, $27$:
$81 = 3^4$

The $6$th heptagonal number after $1$, $7$, $18$, $34$, $55$:
$81 = 1 + 7 + 11 + 16 + 21 + 26 = \dfrac {6 \paren {5 \times 6 - 3} } 2$

The $6$th number after $1$, $3$, $22$, $66$, $70$, and $2$nd square number after $1$, whose $\sigma$ value is square:
$\map \sigma {81} = 121 = 11^2$

The index (after $2$, $3$, $6$, $30$, $75$) of the $6$th Woodall prime:
$81 \times 2^{81} - 1$

The $9$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$:
$81 = 9 \times 9$

The $9$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$ which has no more than $2$ distinct digits

The $13$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$

The $31$st of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $\ldots$

The $35$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{81} = 2 \, 417 \, 851 \, 639 \, 229 \, 258 \, 349 \, 412 \, 352$

The $43$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$, $66$, $67$, $72$, $77$, $80$ which cannot be expressed as the sum of distinct pentagonal numbers

One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:
$81 \to 63 \to 27 \to 45 \to 09 \to 81$

$81 = 9 \times 9 = 9 \times \left({1 + 8}\right)$