A277169 Product of squares of proper divisors of n.

1, 1, 1, 4, 1, 36, 1, 64, 9, 100, 1, 20736, 1, 196, 225, 4096, 1, 104976, 1, 160000, 441, 484, 1, 191102976, 25, 676, 729, 614656, 1, 729000000, 1, 1048576, 1089, 1156, 1225, 78364164096, 1, 1444, 1521, 4096000000, 1, 5489031744, 1, 3748096, 4100625, 2116, 1, 28179280429056, 49, 6250000

Offset: 1

Author

Ilya Gutkovskiy, Oct 19 2016

Keywords

• nonn
• easy

Examples

			a(6) = 36 because 6 has 3 proper divisors {1,2,3} and 1^2*2^2*3^2 = 36.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor Product
• Eric Weisstein's World of Mathematics, Proper divisors

Crossrefs

Cf. A000290, A001248, A007422, A007955, A007956, A008578, A062758.

Programs

Maple

seq(n^(numtheory:-tau(n)-2), n=1..50); # Robert Israel, Nov 13 2016

Mathematica

Table[n^(DivisorSigma[0, n] - 2), {n, 1, 50}]

Formula

a(n) = n^(sigma_0(n)-2).

a(n) = n^A000005(n)/A000290(n).

a(n) = A000290(A007956(n))/A000290(n).

a(n) = A000290(A007955(n)/n)/A000290(n).

a(n) = A062758(n)/A000290(n).

a(n) = 1 if n is prime or n = 1 (A008578).

a(n) = n if n is square of prime (A001248).

a(n) = n^2 if n is multiplicatively perfect number (A007422).