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A368171 a(n) is the smallest divisor d of n such that n/d is a cubefull exponentially odd number (A335988).

%I A368171 #7 Dec 16 2023 09:01:51
%S A368171 1,2,3,4,5,6,7,1,9,10,11,12,13,14,15,2,17,18,19,20,21,22,23,3,25,26,1,
%T A368171 28,29,30,31,1,33,34,35,36,37,38,39,5,41,42,43,44,45,46,47,6,49,50,51,
%U A368171 52,53,2,55,7,57,58,59,60,61,62,63,2,65,66,67,68,69,70
%N A368171 a(n) is the smallest divisor d of n such that n/d is a cubefull exponentially odd number (A335988).
%C A368171 First differs from A050985 at n = 32, and from A367699 at n = 64.
%H A368171 Amiram Eldar, <a href="/A368171/b368171.txt">Table of n, a(n) for n = 1..10000</a>
%F A368171 Multiplicative with a(p^e) = p^e if e <= 2, a(p^e) = 1 if e is odd and e > 1, and p otherwise.
%F A368171 a(n) = n/A368170(n).
%F A368171 a(n) >= 1, with equality if and only if n is in A335988.
%F A368171 a(n) <= n, with equality if and only if n is cubefree (A004709).
%F A368171 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^5 + 1/p^6) = 0.42246686366220037592... .
%t A368171 f[p_, e_] := If[e <= 2, p^e, If[EvenQ[e], p, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368171 (PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2] <= 2, f[i,1]^f[i,2], if(f[i,2]%2, 1, f[i,1])))};
%Y A368171 Cf. A004709, A335988, A368170.
%Y A368171 Cf. A050985, A367699.
%K A368171 nonn,easy,mult
%O A368171 1,2
%A A368171 _Amiram Eldar_, Dec 14 2023