# 35

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## Number

$35$ (thirty-five) is:

$5 \times 7$

The $1$st positive integer $n$ such that $\map \sigma n = \dfrac {\map \phi n \times \map \tau n} 2$:
$\map \sigma {35} = 48 = \dfrac {\map \phi {35} \times \map \tau {35} } 2$

The $4$th pentatope number after $1$, $5$, $15$:
$35 = 1 + 4 + 10 + 20 = \dfrac {4 \paren {4 + 1} \paren {4 + 2} \paren {4 + 3} } {24}$

The $4$th integer after $7$, $13$, $19$ the decimal representation of whose square can be split into two parts which are each themselves square:
$35^2 = 1225$; $1 = 1^2$, $225 = 15^2$

The $5$th pentagonal number after $1$, $5$, $12$, $22$:
$35 = 1 + 4 + 7 + 10 + 13 = \dfrac {5 \paren {3 \times 5 - 1} } 2$

The $5$th tetrahedral number, after $1$, $4$, $10$, $20$:
$35 = 1 + 3 + 6 + 10 + 15 = \dfrac {5 \paren {5 + 1} \paren {5 + 2} } 6$

The $5$th integer $n$ after $1$, $3$, $15$, $30$ with the property that $\map \tau n \divides \map \phi n \divides \map \sigma n$:
$\map \tau {35} = 4$, $\map \phi {35} = 24$, $\map \sigma {35} = 48$

The number of distinct free hexominoes

The $9$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$:
$35 = \dfrac {5 \paren {3 \times 5 - 1} } 2$

The $13$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$:
$35 = 5 \times 7$

The $18$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $35$, $\ldots$

The $19$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

The $25$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
$2^{35} = 34 \, 359 \, 738 \, 368$

The number of pairs of twin primes less than $1000$

The maximum length of a non-crossing knight's tour on a standard chessboard.

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