400

Number
$400$ (four hundred) is:


 * $2^4 \times 5^2$


 * The number of different ways of playing the first $2$ moves in chess


 * The $2$nd after $121$ of the two square numbers which is the sum of consecutive powers of a positive integer:
 * $400 = 7^0 + 7^1 + 7^2 + 7^3$


 * The $6$th even integer after $2$, $4$, $94$, $96$, $98$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes


 * The $8$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$ to be the divisor sum of some (strictly) positive integer:
 * $400 = \map {\sigma_1} {343}$


 * The $18$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $320$, $324$, $336$, $352$, $360$, $384$ with $6$ or more prime factors:
 * $400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5$


 * The $18$th positive integer which cannot be expressed as the sum of a square and a prime:
 * $1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $\ldots$


 * The $20$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $225$, $256$, $289$, $324$, $361$:
 * $400 = 20 \times 20$


 * The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $21$ different ways


 * The $24$th number, and $3$rd square number after $1$, $81$, whose Divisor Sum is square:


 * The $34$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $144$, $169$, $196$, $200$, $216$, $225$, $243$, $256$, $288$, $289$, $324$, $343$, $361$, $392$:
 * $400 = 2^4 \times 5^2$


 * The divisor sum of $7^3$, and so:
 * $400 = 20^2 = 7^0 + 7^1 + 7^2 + 7^3$

Also see

 * Divisor Sum of 343