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A382906 The powerful part of the n-th biquadratefree number.

%I A382906 #9 Apr 09 2025 04:19:29
%S A382906 1,1,1,4,1,1,1,8,9,1,1,4,1,1,1,1,9,1,4,1,1,1,8,25,1,27,4,1,1,1,1,1,1,
%T A382906 36,1,1,1,8,1,1,1,4,9,1,1,49,25,1,4,1,27,1,8,1,1,1,4,1,1,9,1,1,1,4,1,
%U A382906 1,1,72,1,1,25,4,1,1,1,1,1,4,1,1,1,8,1,9,1
%N A382906 The powerful part of the n-th biquadratefree number.
%H A382906 Amiram Eldar, <a href="/A382906/b382906.txt">Table of n, a(n) for n = 1..10000</a>
%H A382906 Vaclav Kotesovec, <a href="/A382906/a382906.jpg">Graph - the asymptotic ratio (100000000 terms)</a>
%F A382906 a(n) = A057521(A046100(n)).
%F A382906 a(n) = A046100(n)/A382905(n).
%F A382906 Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(4)^(3/2) * Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 2.21177275344948791706... .
%t A382906 f[p_, e_] := p^If[e > 1, e, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 4 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
%o A382906 (PARI) list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 4, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] > 1, f[i, 2], 0)), ", ")));}
%Y A382906 Cf. A013662, A046100, A057521.
%Y A382906 Similar sequences: A382902, A382903, A382904, A382905.
%K A382906 nonn,easy
%O A382906 1,4
%A A382906 _Amiram Eldar_, Apr 08 2025