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A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

%I A329376 #24 May 31 2024 08:17:17
%S A329376 1,1,1,2,1,1,1,1,3,1,1,2,1,1,1,1,1,3,1,2,1,1,1,1,5,1,1,2,1,1,1,1,1,1,
%T A329376 1,6,1,1,1,1,1,1,1,2,3,1,1,1,7,5,1,2,1,1,1,1,1,1,1,2,1,1,3,1,1,1,1,2,
%U A329376 1,1,1,3,1,1,5,2,1,1,1,1,1,1,1,2,1,1,1,1,1,3,1,2,1,1,1,1,1,7,3,10,1,1,1,1,1
%N A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.
%C A329376 Product of those distinct prime factors that occur exactly twice in the prime factorization of n, that is, whose exponent is 2.
%H A329376 Antti Karttunen, <a href="/A329376/b329376.txt">Table of n, a(n) for n = 1..65537</a>
%H A329376 Vaclav Kotesovec, <a href="/A329376/a329376.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.
%F A329376 a(n) <= A000196(n).
%F A329376 From _Amiram Eldar_, Feb 11 2023: (Start)
%F A329376 a(n) <= sqrt(n), with equality if and only if n is in A062503.
%F A329376 a(n) = 1 if and only if n is in A337050. (End)
%F A329376 From _Vaclav Kotesovec_, May 31 2024: (Start)
%F A329376 Dirichlet g.f.: zeta(2*s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
%F A329376 Let f(s) = Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).
%F A329376 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
%F A329376 f(1) = Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,
%F A329376 f'(1) = f(1) * Sum_{p prime} (9*p^2 - 12*p + 5) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 3.78385641685861932254178374972226733621783278751462026270346293...
%F A329376 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A329376 f[p_, e_] := If[e == 2, p, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 11 2023 *)
%o A329376 (PARI) A329376(n) = { my(f = factor(n)); prod(i=1,#f~,f[i, 1]^(2 == f[i, 2])); };
%o A329376 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X^2 + X + X^2/(-1 + 1/X)))[n], ", ")) \\ _Vaclav Kotesovec_, May 31 2024
%Y A329376 Cf. A000196, A062503, A337050.
%Y A329376 Row 3 of array A106177, and the square roots of its row 9.
%K A329376 nonn,easy,mult
%O A329376 1,4
%A A329376 _Antti Karttunen_, Nov 16 2019