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A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

%I A356192 #17 Nov 13 2022 08:40:51
%S A356192 1,8,27,8,125,216,343,8,27,1000,1331,216,2197,2744,3375,32,4913,216,
%T A356192 6859,1000,9261,10648,12167,216,125,17576,27,2744,24389,27000,29791,
%U A356192 32,35937,39304,42875,216,50653,54872,59319,1000,68921,74088,79507,10648,3375,97336
%N A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.
%C A356192 First differs from A053149 and A356193 at n=16.
%H A356192 Amiram Eldar, <a href="/A356192/b356192.txt">Table of n, a(n) for n = 1..10000</a>
%F A356192 Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
%F A356192 a(n) = n iff n is in A335988.
%F A356192 a(n) = A356191(n) iff n is a powerful number (A001694).
%F A356192 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
%F A356192 Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - _Amiram Eldar_, Nov 13 2022
%t A356192 f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
%o A356192 (PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};
%Y A356192 Cf. A001694, A013664, A268335, A335988.
%Y A356192 Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.
%K A356192 nonn,easy,mult
%O A356192 1,2
%A A356192 _Amiram Eldar_, Jul 29 2022