A183102 a(n) = product of powerful divisors d of n.

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 4, 1, 1, 1, 512, 1, 9, 1, 4, 1, 1, 1, 32, 25, 1, 243, 4, 1, 1, 1, 16384, 1, 1, 1, 1296, 1, 1, 1, 32, 1, 1, 1, 4, 9, 1, 1, 512, 49, 25, 1, 4, 1, 243, 1, 32, 1, 1, 1, 4, 1, 1, 9, 1048576, 1, 1, 1, 4, 1, 1, 1, 746496, 1

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Comments

a(n) = product of divisors d of n from set A001694 - powerful numbers.

Sequence is not the same as A183104(n): a(72) = 746496, A183104(72) = 10368.

Not multiplicative: a(4)*a(9) = 4*9=36 <> a(36) = 1296. - R. J. Mathar, Jun 07 2011

Examples

			For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.
		

Links

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

Crossrefs

Cf. A001694, A007955, A183097, A183103, A183104.

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:
A183102 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001694(d) then a := a*d; end if; end do; a ; end proc:
seq(A183102(n),n=1..70) ; # 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

A183102(n) = { my(m=1); fordiv(n, d, if(ispowerful(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A007955(n) / A183103(n).

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