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单词 ProductOfDivisorsFunction
释义

product of divisors function


The product of all positive divisorsMathworldPlanetmathPlanetmath of a nonzero integer n is equal nτ(n), where tau function τ(n) expresses the number of the positive divisors of n.

Proof.  Let  t=τ(n)  and the positive divisors of n be a1<a2<<at.

If n is not a square of an integer, t is even (see http://planetmath.org/node/11781parity of τ function), whence

{a1at=na2at-1=nat2at+22=n.

Thus

dnd=a1a2at=nt2.

If n is a square of an integer, t is odd, and we have

{a1at=na2at-1=nat-12at+32=nat+12=n12.

In this case we obtain a result:

dnd=a1a2at=nt-12+12=nt2

Note.  The absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the product of all divisors is nτ(n).

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更新时间:2025/5/4 8:11:10