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A370256 The number of ways in which n can be expressed as b^2 * c^3, with b and c >= 1.

%I A370256 #45 Apr 15 2025 08:55:08
%S A370256 1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,
%T A370256 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,
%U A370256 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0
%N A370256 The number of ways in which n can be expressed as b^2 * c^3, with b and c >= 1.
%C A370256 First differs from A075802 and A112526 at n = 64.
%C A370256 The least number k such that a(k) = n is A005179(n)^6.
%C A370256 The indices of records are the sixth powers of the highly composite numbers, A002182(n)^6.
%H A370256 Amiram Eldar, <a href="/A370256/b370256.txt">Table of n, a(n) for n = 1..10000</a>
%F A370256 Multiplicative with a(p^e) = A103221(e).
%F A370256 a(n) > 0 if and only if n is a powerful number (A001694).
%F A370256 a(A001694(n)) = A057523(n).
%F A370256 a(n^6) = A000005(n).
%F A370256 Sum_{k=1..n} a(k) ~ zeta(3/2) * sqrt(n) + zeta(2/3) * n^(1/3).
%F A370256 Dirichlet generating function: zeta(2*s)*zeta(3*s). - _Vaclav Kotesovec_, Feb 23 2024
%e A370256 1 = 1^2 * 1^3, so a(1) = 1.
%e A370256 64 = 1^2 * 4^3 = 8^2 * 1^3, so a(64) = 2.
%e A370256 4096 = 64^2 * 1^3 = 8^2 * 4^3 = 1^2 * 16^3, so a(4096)= 3.
%t A370256 f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A370256 (PARI) a(n) = vecprod(apply(x -> (x+2)\2 - (x+2)\3, factor(n)[, 2]));
%o A370256 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X^2)*(1 - X^3)))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 23 2024
%o A370256 (Python)
%o A370256 from math import prod
%o A370256 from sympy import factorint
%o A370256 def A370256(n): return prod((e>>1)+1-(e+2)//3 for e in factorint(n).values()) # _Chai Wah Wu_, Apr 15 2025
%Y A370256 Cf. A000005, A001694, A002182, A005179, A057523, A075802, A078434, A103221, A112526.
%K A370256 nonn,easy,mult
%O A370256 1,64
%A A370256 _Amiram Eldar_, Feb 23 2024