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A355264 a(n) = n * largest-nth-power(n, 2) = n * A000188(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.

%I A355264 #18 Sep 21 2023 01:45:51
%S A355264 1,2,3,8,5,6,7,16,27,10,11,24,13,14,15,64,17,54,19,40,21,22,23,48,125,
%T A355264 26,81,56,29,30,31,128,33,34,35,216,37,38,39,80,41,42,43,88,135,46,47,
%U A355264 192,343,250,51,104,53,162,55,112,57,58,59,120,61,62,189,512
%N A355264 a(n) = n * largest-nth-power(n, 2) = n * A000188(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.
%H A355264 Michael De Vlieger, <a href="/A355264/b355264.txt">Table of n, a(n) for n = 1..10000</a>
%F A355264 Multiplicative with a(p^e) = p^(e+floor(e/2)). - _Amiram Eldar_, Jul 13 2022
%F A355264 From _Amiram Eldar_, Sep 21 2023: (Start)
%F A355264 Dirichlet g.f.: zeta(s-1) * zeta(2*s-3)/ zeta(2*s-2).
%F A355264 Sum_{k=1..n} a(k) ~ (3*n^2/(4*Pi^2)) * (2*log(n) + 6*gamma - 4*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620). (End)
%p A355264 with(NumberTheory): seq(n*LargestNthPower(n, 2), n = 1..64);
%t A355264 Table[n*Times @@ (#1^Floor[#2/2] & @@@ FactorInteger[n]), {n, 64}] (* _Michael De Vlieger_, Jul 12 2022 *)
%o A355264 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + f[i,2]\2));} \\ _Amiram Eldar_, Sep 21 2023
%Y A355264 Cf. A000027, A000188, A007913.
%Y A355264 Cf. A001620, A306016.
%K A355264 nonn,easy,mult
%O A355264 1,2
%A A355264 _Peter Luschny_, Jul 12 2022