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A183104 a(n) = product of divisors of n that are perfect powers.

%I A183104 #18 Mar 07 2024 12:01:00
%S A183104 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 A183104 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 A183104 1,4,1,1,9,1048576,1,1,1,4,1,1,1,10368
%N A183104 a(n) = product of divisors of n that are perfect powers.
%C A183104 Sequence is not the same as A183102: a(72) = 10368, A183102(72) = 746496.
%C A183104 Not multiplicative, as a(4)*a(9) <> a(36). - _R. J. Mathar_, Jun 07 2011
%H A183104 Antti Karttunen, <a href="/A183104/b183104.txt">Table of n, a(n) for n = 1..16385</a>
%F A183104 a(n) = A007955(n) / A183105(n).
%F A183104 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 A183104 For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.
%p A183104 isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
%p A183104 A183104 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001597(d) then a := a*d; end if; end do; a ; end proc:
%p A183104 seq(A183104(n),n=1..72) ; # _R. J. Mathar_, Jun 07 2011
%t A183104 perfPQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Table[Times@@Select[Divisors[n],perfPQ[#]&],{n,120}] (* _Harvey P. Dale_, Mar 07 2024 *)
%o A183104 (PARI) A183104(n) = { my(m=1); fordiv(n, d, if(ispower(d), m *= d)); m; }; \\ _Antti Karttunen_, Oct 07 2017
%Y A183104 Cf. A001597, A007955, A091051, A183102, A183105.
%K A183104 nonn
%O A183104 1,4
%A A183104 _Jaroslav Krizek_, Dec 25 2010