A183103 a(n) = product of non-powerful divisors d of n.

1, 2, 3, 2, 5, 36, 7, 2, 3, 100, 11, 432, 13, 196, 225, 2, 17, 648, 19, 2000, 441, 484, 23, 10368, 5, 676, 3, 5488, 29, 810000, 31, 2, 1089, 1156, 1225, 7776, 37, 1444, 1521, 80000, 41, 3111696, 43, 21296, 10125, 2116, 47, 497664, 7, 5000, 2601, 35152, 53, 34992, 3025, 307328, 3249, 3364, 59, 11664000000, 61, 3844, 27783, 2, 4225, 18974736, 67, 78608, 4761, 24010000, 71, 186624, 73

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Comments

Sequence is not the same as A183105: a(72) = 186624, A183105(72) = 13436928.

Not multiplicative, for example a(2)*a(3) <> a(6). - R. J. Mathar, Jun 07 2011

Examples

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 1*2*3*6*12 = 432.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385

Crossrefs

Cf. A001694, A007955, A183098, A183102, A183105.

Programs

Maple

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A183103 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if not isA001694(d) then a := a*d; end if; end do; a ; end proc:
seq(A183103(n),n=1..73) ; # R. J. Mathar, Jun 07 2011

Mathematica

powerfulQ[n_] := Min[FactorInteger[n][[All, 2]]] > 1;
a[n_] := Times @@ Select[Divisors[n], !powerfulQ[#]&];
Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jun 01 2024 *)

PARI

A183103(n) = { my(m=1); fordiv(n, d, if(!ispowerful(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A007955(n) / A183102(n).

a(1) = 1, a(p) = p, a(pq) = (pq)^2, a(pq...z) = (pq...z)^(2^(k-1)), a(p^k) = p, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.