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

1,3

Comments

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

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

Links

Table of n, a(n) for n=1..58.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A334420, A334421.

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

Sequence in context: A369121 A347556 A368340 * A104053 A187369 A039512

Adjacent sequences: A338169 A338170 A338171 * A338173 A338174 A338175

Keyword

nonn

Author

Jaroslav Krizek, Oct 14 2020

Status

approved