# 34

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## Number

$34$ (thirty-four) is:

$2 \times 17$

The $4$th heptagonal number after $1$, $7$, $18$:
$34 = 1 + 7 + 11 + 16 = \dfrac {4 \left({5 \times 4 - 3}\right)} 2$

The $12$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$:
$34 = 2 \times 17$

The $9$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$:
$34 = 13 + 21$

The $3$rd nontotient after $14$, $26$:
$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 34$
where $\phi \left({m}\right)$ denotes the Euler $\phi$ function

The $3$rd noncototient after $10$, $26$:
$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 34$
where $\phi \left({m}\right)$ denotes the Euler $\phi$ function

The smallest positive integer to be the sum of $2$ lucky numbers in $4$ different ways:
$34 = 1 + 33 = 3 + 31 = 9 + 25 = 13 + 21$

The $24$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{34} = 17 \, 179 \, 869 \, 184$

The $1$st positive integer which can be expressed as the sum of $2$ odd primes in $4$ ways:
$34 = 31 + 3 = 29 + 7 = 23 + 11 = 17 + 17$

The $4$th positive integer which cannot be expressed as the sum of a square and a prime:
$1$, $10$, $25$, $34$, $\ldots$

The $2$nd non-square positive integer which cannot be expressed as the sum of a square and a prime:
$10$, $34$, $\ldots$

The magic constant of a magic square of order $4$, after $1$, $(5)$, $15$:
$34 = \displaystyle \dfrac 1 4 \sum_{k \mathop = 1}^{4^2} k = \dfrac {4 \paren {4^2 + 1} } 2$