# 43

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## Number

$43$ (forty-three) is:

The $14$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$

The index of the $1$st non-integer term of Göbel's sequence.

The lower end of the $4$th record-breaking gap between twin primes:
$59 - 43 = 16$

The $5$th of $11$ primes of the form $2 x^2 + 11$:
$2 \times 4^2 + 11 = 43$ (Previous  ... Next)

The $6$th prime $p$ after $11$, $23$, $29$, $37$, $41$ such that the Mersenne number $2^p - 1$ is composite

The $2$nd of the $6$th pair of twin primes, with $41$

The $12$th lucky number:
$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $\ldots$

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

The $19$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $\ldots$

The $21$st odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $\ldots$

The $26$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$, $37$, $38$, $42$ which cannot be expressed as the sum of distinct pentagonal numbers.