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%I #13 Feb 23 2024 07:10:37
%S 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,
%T 1,1,1,2,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1
%N Number of ways n-th powerful (1) number can be expressed as b^2*c^3.
%C Always positive. (Would be zero for any non-powerful number.)
%C The positive terms of A370256. - _Amiram Eldar_, Feb 23 2024
%H Amiram Eldar, <a href="/A057523/b057523.txt">Table of n, a(n) for n = 1..10000</a>
%F From _Amiram Eldar_, Feb 23 2024: (Start)
%F a(n) = A370256(A001694(n)).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3). (End)
%e a(40) = 2 since the 40th powerful number is 576 = 24^2*1^3 = 3^2*4^3.
%t f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10000], # > 0 &] (* _Amiram Eldar_, Feb 23 2024 *)
%o (PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(apply(x -> (x+2)\2 - (x+2)\3, e)), ", ")));} \\ _Amiram Eldar_, Feb 23 2024
%Y Cf. A001694, A002117, A370256.
%K nonn,easy
%O 1,11
%A _Henry Bottomley_, Sep 02 2000
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