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A382903 The largest cubefree unitary divisor of the n-th biquadratefree number.

%I A382903 #7 Apr 09 2025 04:19:43
%S A382903 1,2,3,4,5,6,7,1,9,10,11,12,13,14,15,17,18,19,20,21,22,23,3,25,26,1,
%T A382903 28,29,30,31,33,34,35,36,37,38,39,5,41,42,43,44,45,46,47,49,50,51,52,
%U A382903 53,2,55,7,57,58,59,60,61,62,63,65,66,67,68,69,70,71,9,73
%N A382903 The largest cubefree unitary divisor of the n-th biquadratefree number.
%H A382903 Amiram Eldar, <a href="/A382903/b382903.txt">Table of n, a(n) for n = 1..10000</a>
%F A382903 a(n) = A360539(A046100(n)).
%F A382903 a(n) = A382902(n)^3 / A046100(n)^2.
%F A382903 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)^2 * Product_{p prime} (1 - 1/p^3 + 1/p^6 - 1/p^7) = 0.98420942697128846925... .
%t A382903 f[p_, e_] := p^If[e < 3, e, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 4 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
%o A382903 (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] < 3, f[i, 2], 0)), ", ")));}
%Y A382903 Cf. A046100, A013662, A360539.
%Y A382903 Similar sequences: A382902, A382904, A382905, A382906.
%K A382903 nonn,easy
%O A382903 1,2
%A A382903 _Amiram Eldar_, Apr 08 2025