# 33

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## Number

$33$ (thirty-three) is:

$3 \times 11$

The $1$st of the smallest triple of $3$ consecutive semiprimes:
$33$, $34$, $35$

The smallest natural number solution to $\map \sigma n = \map \sigma {n + 2}$:
$\map \sigma {33} = 48 = \map \sigma {35}$

The sum of the first $4$ factorials:
$33 = 1 + 2 + 6 + 24 = 1! + 2! + 3! + 4!$

The $5$th palindromic lucky number:
$1$, $3$, $7$, $9$, $33$, $\ldots$

The $6$th (and last) integer after $2$, $5$, $8$, $12$, $23$ which cannot be expressed as the sum of distinct triangular numbers

The $7$th integer after $0$, $1$, $3$, $5$, $7$, $9$ which is palindromic in both decimal and binary:
$33_{10} = 100 \, 001_2$

The $9$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$

The $10$th lucky number:
$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $\ldots$

The $11$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$:
$33 = 3 \times 11$

The $11$th, and probably largest, integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $18$ such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:
$2^{33} = 8 \, 589 \, 934 \, 592$, $5^{33} = 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$

The $12$th and largest (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$, $12$, $15$, $18$ which cannot be expressed as the sum of exactly $5$ non-zero squares.

The $15$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$, $17$, $18$, $30$ such that $5^n$ contains no zero in its decimal representation:
$5^{33} = 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$

The $17$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $\ldots$

The $18$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $\ldots$

The $22$nd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$ which cannot be expressed as the sum of distinct pentagonal numbers

The $23$rd integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{33} = 8 \, 589 \, 934 \, 592$