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

%I A183102 #22 Jun 01 2024 03:08:16
%S A183102 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,
%T A183102 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,
%U A183102 1,4,1,1,9,1048576,1,1,1,4,1,1,1,746496,1
%N A183102 a(n) = product of powerful divisors d of n.
%C A183102 a(n) = product of divisors d of n from set A001694 - powerful numbers.
%C A183102 Sequence is not the same as A183104(n): a(72) = 746496, A183104(72) = 10368.
%C A183102 Not multiplicative: a(4)*a(9) = 4*9=36 <> a(36) = 1296. - _R. J. Mathar_, Jun 07 2011
%H A183102 Antti Karttunen, <a href="/A183102/b183102.txt">Table of n, a(n) for n = 1..16385</a>
%F A183102 a(n) = A007955(n) / A183103(n).
%F A183102 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.
%e A183102 For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.
%p A183102 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:
%p A183102 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:
%p A183102 seq(A183102(n),n=1..70) ; # _R. J. Mathar_, Jun 07 2011
%t A183102 powerfulQ[n_] := Min[FactorInteger[n][[All, 2]]] > 1;
%t A183102 a[n_] := Times @@ Select[Divisors[n], powerfulQ];
%t A183102 Table[a[n], {n, 1, 73}] (* _Jean-François Alcover_, Jun 01 2024 *)
%o A183102 (PARI) A183102(n) = { my(m=1); fordiv(n, d, if(ispowerful(d), m *= d)); m; }; \\ _Antti Karttunen_, Oct 07 2017
%Y A183102 Cf. A001694, A007955, A183097, A183103, A183104.
%K A183102 nonn
%O A183102 1,4
%A A183102 _Jaroslav Krizek_, Dec 25 2010