# 61

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## Number

$61$ (sixty-one) is:

The $18$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$

The $1$st of the $1$st ordered quadruple of consecutive integers that have sigma values which are strictly increasing:
$\map \sigma {61} = 62$, $\map \sigma {62} = 96$, $\map \sigma {63} = 104$, $\map \sigma {64} = 127$

The $1$st positive integer whose reciprocal, when expressed in decimal notation, contains an equal number ($6$) of each of the digits from $0$ to $9$:
$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

The $3$rd after $21$, $29$ of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number, for example:
$1 \, 318 \, 820 \, 881 = 1 \, 739 \, 288 \, 516 \, 161 \, 616 \, 161$

The $5$th Keith number after $14$, $19$, $28$, $47$:
$6$, $1$, $7$, $8$, $15$, $23$, $38$, $61$, $\ldots$

The $5$th centered hexagonal number after $1$, $7$, $19$, $37$:
$61 = 1 + 6 + 12 + 18 + 24 = 5^3 - 4^3$

The $5$th of $29$ primes of the form $2 x^2 + 29$:
$2 \times 4^2 + 29 = 61$ (Previous  ... Next)

The $6$th of $11$ primes of the form $2 x^2 + 11$:
$2 \times 5^2 + 11 = 61$ (Previous  ... Next)

The $2$nd of the $7$th pair of twin primes, with $59$

The $8$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$ such that the Fibonacci number $F_n$ ends in $n$

The $8$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$:
$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

The index of the $9$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$:
$M_{61} = 2^{61} - 1 = 2 \, 305 \, 843 \, 009 \, 213 \, 693 \, 951$

The $12$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$ such that no factorial of an integer can end with $n$ zeroes

The $12$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$ such that $m = \displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime

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

The $28$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $\ldots$

The $36$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $44$, $45$, $46$, $49$, $50$, $54$, $55$, $59$, $60$ which cannot be expressed as the sum of distinct pentagonal numbers