# 144

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## Number

$144$ (one hundred and forty-four) is:

$2^4 \times 3^2$

The $2$nd power of $12$ after $(1)$, $12$:
$144 = 12^2$

The $3$rd and last Square Fibonacci Number after $0$ and $1$
$144 = F_{12} = 55 + 89 = 12^2$

The $3$rd and last Fibonacci number after $0$, $1$ which equals the square of its index
$144 = F_{12} = 12^2 = 55 + 89$

The $4$th positive integer after $64$, $96$, $128$ with $6$ or more prime factors:
$144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$

The $5$th square number after $1$, $4$, $36$, $121$ to be the $\sigma$ (sigma) value of some (strictly) positive integer:
$144 = \map \sigma {66} = \map \sigma {70} = \map \sigma {94} = \map \sigma {115} = \map \sigma {119}$

The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $6$ different ways

The $11$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $121$ which has no more than $2$ distinct digits

The smallest positive integer which can be expressed as the sum of $2$ odd primes in $11$ ways.

The $12$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$:
$144 = 55 + 89$

The $12$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$:
$144 = 12 \times 12 = \paren {2^2 \times 3}^2$
Hence in duodecimal notation:
$100$

The $20$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$, $121$, $125$, $128$

The $21$st Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$, $132$, $135$:
$144 = 9 \times 16 = 9 \times \paren {1 \times 4 \times 4}$

The $24$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$:
$\map \sigma {144} = 403$

## Historical Note

The word gross, now almost obsolete in this context, means a set of $144$.