# 66

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## Number

$66$ (sixty-six) is:

$2 \times 3 \times 11$

The $2$nd after $55$ of the $3$ repdigit numbers which are also triangular

The $3$rd sphenic number after $30$, $42$:
$66 = 2 \times 3 \times 11$

The $3$rd hexamorphic number after $1$, $45$:
$66 = H_6$

The $4$th number after $1$, $3$, $22$ whose $\sigma$ value is square:
$\map \sigma {66} = 144 = 12^2$

The $5$th palindromic triangular number after $0$, $1$, $3$, $6$ whose index is itself palindromic:
$66 = T_{11}$

The $6$th hexagonal number after $1$, $6$, $15$, $28$, $45$:
$66 = 1 + 5 + 9 + 13 + 17 + 21 = 6 \paren {2 \times 6 - 1}$

The $6$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$
$66 = \dfrac {11 \times \paren {11 + 1} } 2$
The $2$nd of those after $55$ which can be split into two palindromic halves

The $11$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$:
$66 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = \dfrac {11 \times \paren {11 + 1} } 2$

The $15$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$:
$66 = 11 + 22 + 33$

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

The $38$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$ which cannot be expressed as the sum of distinct pentagonal numbers