90

Number
$90$ (ninety) is:


 * $2 \times 3^2 \times 5$


 * The $3$rd unitary perfect number after $6$, $60$:
 * $90 = 1 + 2 + 5 + 9 + 10 + 18 + 45$


 * The $4$th element of the $1$st set of $4$ positive integers which form an arithmetic sequence which all have the same Euler $\phi$ value:
 * $\map \phi {72} = \map \phi {78} = \map \phi {84} = \map \phi {90} = 24$


 * The smallest positive integer which can be expressed as the sum of $2$ odd primes in $9$ ways.


 * The $11$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$:
 * $\nexists m \in \Z_{>0}: \map \phi m = 90$
 * where $\map \phi n$ denotes the Euler $\phi$ function


 * The $20$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$:
 * $\map \sigma {90} = 234$


 * The $20$th of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
 * $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$, $42$, $48$, $60$, $90$, $\ldots$


 * The $21$st semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$, $84$, $88$:
 * $90 = 15 + 30 + 45$


 * The $35$th and last of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
 * $3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $82$, $87$, $88$, $90$