40

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Number

$40$ (forty) is:

$2^3 \times 5$

The only number whose name in the English language has all its letters in alphabetical order:

forty

The $4$th octagonal number, after $1$, $8$, $21$:

$40 = 1 + 7 + 13 + 19 = 4 \paren {3 \times 4 - 2}$

The $4$th pentagonal pyramidal number after $1$, $6$, $18$:

$40 = 1 + 5 + 12 + 22 = \dfrac {4^2 \paren {4 + 1} } 2$

The $5$th second pentagonal number after $2$, $7$, $15$, $26$:

$40 = \dfrac {5 \paren {3 \times 5 + 1} } 2$

The $7$th abundant number after $12, 18, 20, 24, 30, 36$:

$1 + 2 + 4 + 5 + 8 + 10 + 20 = 50 > 40$

The $9$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$:

$40 = 2 + 8 + 10 + 20$

The $10$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$:

$40 = \dfrac {5 \paren {3 \times 5 + 1} } 2$

The $16$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:

$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $\ldots$

The $19$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$, $27$, $30$, $36$:

$40 = 10 \times 4 = 10 \times \paren {4 + 0}$

Also see

• Previous  ... Next: Pentagonal Pyramidal Number

• Previous  ... Next: Octagonal Number

• Previous  ... Next: Second Pentagonal Number

• Previous  ... Next: Generalized Pentagonal Number

• Previous  ... Next: Abundant Number
• Previous  ... Next: Semiperfect Number
• Previous  ... Next: Harshad Number

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

Historical Note

In the Bible, $40$ was frequently used as a number associated with a long period of time:

The Israelites wandered for $40$ years in the Wilderness.

Also in the Wilderness, Jesus fasted for $40$ days and $40$ nights.

There are $40$ rods, poles or perches in a furlong.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $40$