90

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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 $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$:

$\map {\sigma_1} {90} = 234$

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:

$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $68$, $69$, $74$, $75$, $79$, $81$, $82$, $87$, $88$, $90$

The $48$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

Arithmetic Functions on $90$

\(\ds \map {\sigma_0} { 90 }\)

\(=\)

\(\ds 12\)

$\sigma_0$ of $90$

\(\ds \map \phi { 90 }\)

\(=\)

\(\ds 24\)

$\phi$ of $90$

\(\ds \map {\sigma_1} { 90 }\)

\(=\)

\(\ds 234\)

$\sigma_1$ of $90$

Also see

• Previous  ... Next: Integers whose Number of Representations as Sum of Two Primes is Maximum
• Previous  ... Next: Unitary Perfect Number

• Previous  ... Next: Integers not Expressible as Sum of Distinct Primes of form 6n-1
• Previous  ... Next: Highly Abundant Number
• Previous  ... Next: Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways
• Previous  ... Next: 4 Positive Integers in Arithmetic Sequence which have Same Euler Phi Value

• Previous  ... Next: Nontotient

• Previous: 91 is Pseudoprime to 35 Bases less than 91
• Previous  ... Next: Semiperfect Number

Historical Note

There are $90$ degrees in a right angle.