# 36

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## Number

$36$ (thirty-six) is:

$2^2 \times 3^2$

The sum of the first $3$ cubes:
$36 = 1^3 + 2^3 + 3^3$

The $2$nd power of $6$ after $(1)$, $6$:
$36 = 6^2$

The $2$nd number after $1$ to be both square and triangular:
$36 = 6^2 = \dfrac {8 \times \paren {8 + 1} } 2$

The $3$rd of three $2$-digit integers divisible by both the sum and product of its digits:
$36 = \paren {3 + 6} \times 4 = \paren {3 \times 6} \times 2$

The $6$th square number after $1$, $4$, $9$, $16$, $25$:
$36 = 6 \times 6$

The $6$th square after $1$, $4$, $9$, $16$, $25$ which has no more than $2$ distinct digits

The $6$th abundant number after $12$, $18$, $20$, $24$, $30$:
$1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36$

The $7$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$ such that no factorial of an integer can end with $n$ zeroes.

The $7$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$:
$\dfrac {\map \sigma {36} } {36} = \dfrac {91} {36} = 2 \cdotp 52 \dot 7$

The $7$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$:
$\map \tau {36} = 9$

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

The $8$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$:
$36 = 3 + 6 + 9 + 18$

The $9$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$

The $13$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$:
$36 = 8 + 28$

The $14$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$:
$\map \sigma {36} = 91$

The $14$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$:
$36 = 2 \times 18 = 2 \times \left({3 \times 6}\right)$

The $14$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $\ldots$

The $18$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$, $27$, $30$:
$36 = 4 \times 9 = 4 \times \paren {3 + 6}$

The $20$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$, $35$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

The $26$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{36} = 68 \, 719 \, 476 \, 736$

The total of all the entries in a magic cube of order $2$ (if it were to exist), after $1$:
$36 = \displaystyle \sum_{k \mathop = 1}^{2^3} k = \dfrac {2^3 \paren {2^3 + 1} } 2$

$35$ and $4374$ have the same prime factors between them as $36$ and $4375$:
$35 = 5 \times 7$, $4374 = 2 \times 3^7$; $36 = 2^2 \times 3^2$, $4375 = 5^4 \times 7$

## Historical Note

The number $36$ was held in particularly high regard by the ancient Greeks, as it is the sum of the first $4$ even numbers and the first $4$ odd numbers.