101

Number
$101$ (one hundred and one) is:


 * The $26$th prime number


 * The $3$rd prime number after $5$, $19$ of the form $\ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
 * $101 = 5! - 4! + 3! - 2! + 1!$


 * The $4$th unique period prime after $3$, $11$, $37$: its period is $4$:
 * $\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$


 * The smallest positive integer the decimal expansion of whose reciprocal has a period of $4$:
 * $\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$


 * The upper end of the $5$th record-breaking gap between twin primes:
 * $101 - 73 = 28$


 * The $5$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$:
 * $101 = 10^2 + 1$


 * The $5$th positive integer after $1$, $2$, $7$, $11$ whose cube is palindromic:
 * $101^3 = 1 \, 030 \, 301$


 * The $6$th palindromic prime after $2$, $3$, $5$, $7$, $11$


 * The $7$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$ whose square is also palindromic integer
 * $101^2 = 10 \, 201$


 * The $7$th of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 6^2 + 29 = 101$


 * The smaller of the $9$th pair of twin primes, with $103$


 * The $12$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$ such that the Fibonacci number $F_n$ ends in $n$


 * The number of integer partitions for $13$:
 * $\map p {13} = 101$


 * The $36$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $\ldots$


 * The $50$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$, $100$ which cannot be expressed as the sum of distinct pentagonal numbers

Also see