81

Number
$81$ (eighty-one) is:


 * $3^4$


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


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


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


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


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


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


 * The $6$th number after $1, 3, 22, 66, 70$, and $2$nd square number after $1$, whose $\sigma$ value is square:


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


 * 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.

Also see

 * Square Numbers whose Sigma is Square