A338172 a(n) is the product of those divisors d of n such that tau(d) divides sigma(d).

1, 1, 3, 1, 5, 18, 7, 1, 3, 5, 11, 18, 13, 98, 225, 1, 17, 18, 19, 100, 441, 242, 23, 18, 5, 13, 81, 98, 29, 40500, 31, 1, 1089, 17, 1225, 18, 37, 722, 1521, 100, 41, 1555848, 43, 10648, 10125, 1058, 47, 18, 343, 5, 2601, 13, 53, 26244, 3025, 5488, 3249, 29

Offset: 1

Author

Jaroslav Krizek, Oct 14 2020

Keywords

• nonn

Comments

a(n) is the product of arithmetic divisors d of n.

a(n) = pod(n) = A007955(n) for numbers n from A334420.

Examples

			a(6) = 18 because there are 3 arithmetic divisors of 6 (1, 3 and 6): sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3. Product of this divisors is 18.
		

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).

Cf. A334420, A334421.

See A338170 and A338171 for number and sum of such divisors.

Programs

Magma

[&*[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

Mathematica

a[n_] := Times @@ Select[Divisors[n],  Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, if (sigma(d[k]) % numdiv(d[k]), 1, d[k])); \\ Michel Marcus, Oct 15 2020

Formula

a(p) = p for odd primes p (A065091).