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A382905 The powerfree part of the n-th biquadratefree number.

%I A382905 #7 Apr 09 2025 04:19:39
%S A382905 1,2,3,1,5,6,7,1,1,10,11,3,13,14,15,17,2,19,5,21,22,23,3,1,26,1,7,29,
%T A382905 30,31,33,34,35,1,37,38,39,5,41,42,43,11,5,46,47,1,2,51,13,53,2,55,7,
%U A382905 57,58,59,15,61,62,7,65,66,67,17,69,70,71,1,73,74,3,19
%N A382905 The powerfree part of the n-th biquadratefree number.
%H A382905 Amiram Eldar, <a href="/A382905/b382905.txt">Table of n, a(n) for n = 1..10000</a>
%F A382905 a(n) = A055231(A046100(n)).
%F A382905 a(n) = A046100(n)/A382906(n).
%F A382905 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^5 + 1/p^6 - 1/p^7) = 0.75836309584506672244... .
%t A382905 f[p_, e_] := p^If[e < 2, 1, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 4 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
%o A382905 (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] < 2, f[i, 2], 0)), ", ")));}
%Y A382905 Cf. A013662, A046100, A055231.
%Y A382905 Similar sequences: A382902, A382903, A382904, A382906.
%K A382905 nonn,easy
%O A382905 1,2
%A A382905 _Amiram Eldar_, Apr 08 2025