A183103 - OEIS

%I #23 Jun 01 2024 03:08:13

%S 1,2,3,2,5,36,7,2,3,100,11,432,13,196,225,2,17,648,19,2000,441,484,23,

%T 10368,5,676,3,5488,29,810000,31,2,1089,1156,1225,7776,37,1444,1521,

%U 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

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

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

%C Not multiplicative, for example a(2)*a(3) <> a(6). - _R. J. Mathar_, Jun 07 2011

%H Antti Karttunen, <a href="/A183103/b183103.txt">Table of n, a(n) for n = 1..16385</a>

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

%F 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.

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

%p 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 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:

%p seq(A183103(n),n=1..73) ; # _R. J. Mathar_, Jun 07 2011

%t powerfulQ[n_] := Min[FactorInteger[n][[All, 2]]] > 1;

%t a[n_] := Times @@ Select[Divisors[n], !powerfulQ[#]&];

%t Table[a[n], {n, 1, 73}] (* _Jean-François Alcover_, Jun 01 2024 *)

%o (PARI) A183103(n) = { my(m=1); fordiv(n, d, if(!ispowerful(d), m *= d)); m; }; \\ _Antti Karttunen_, Oct 07 2017

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

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Dec 25 2010