84

Number
$84$ (eighty-four) is:


 * $2^2 \times 3 \times 7$


 * The $3$rd 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 $5$th and last after $21$, $29$, $61$, $69$ of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number


 * The $5$th inconsummate number after $62$, $63$, $65$, $75$:
 * $\nexists n \in \Z_{>0}: n = 84 \times \map {s_{10} } n$


 * The $7$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$:
 * $84 = 1 + 3 + 6 + 10 + 15 + 21 + 28 = \dfrac {7 \paren {7 + 1} \paren {7 + 2} } 6$


 * The smallest positive integer which can be expressed as the sum of $2$ odd primes in $8$ ways.


 * The $19$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$:
 * $\map {\sigma_1} {84} = 224$


 * The $19$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$:
 * $84 = 14 + 28 + 42$


 * The $44$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$ which cannot be expressed as the sum of distinct pentagonal numbers.


 * The $47$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$