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A368170 The largest cubefull exponentially odd divisor of n.

%I A368170 #8 Dec 16 2023 09:01:47
%S A368170 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,8,1,1,27,1,1,1,1,32,1,
%T A368170 1,1,1,1,1,1,8,1,1,1,1,1,1,1,8,1,1,1,1,1,27,1,8,1,1,1,1,1,1,1,32,1,1,
%U A368170 1,1,1,1,1,8,1,1,1,1,1,1,1,8,27,1,1,1,1,1
%N A368170 The largest cubefull exponentially odd divisor of n.
%C A368170 First differs from A008834 at n = 32, and from A366906 at n = 64.
%H A368170 Amiram Eldar, <a href="/A368170/b368170.txt">Table of n, a(n) for n = 1..10000</a>
%F A368170 Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = p^e if e is odd and e > 1, and p^(e-1) otherwise.
%F A368170 a(n) = n/A368171(n).
%F A368170 a(n) >= 1, with equality if and only if n is cubefree (A004709).
%F A368170 a(n) <= n, with equality if and only if n is in A335988.
%t A368170 f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], p^(e-1), p^e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368170 (PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2] <= 2, 1, if(!(f[i,2]%2), f[i,1]^(f[i, 2]-1), f[i, 1]^f[i, 2])))};
%Y A368170 Cf. A004709, A335988, A350390, A356192, A368167, A368171.
%Y A368170 Cf. A008834, A366906.
%K A368170 nonn,easy,mult
%O A368170 1,8
%A A368170 _Amiram Eldar_, Dec 14 2023