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A384051 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cubefull number.

%I A384051 #10 May 21 2025 01:33:29
%S A384051 1,1,2,3,4,2,6,8,8,4,10,6,12,6,8,16,16,8,18,12,12,10,22,16,24,12,27,
%T A384051 18,28,8,30,32,20,16,24,24,36,18,24,32,40,12,42,30,32,22,46,32,48,24,
%U A384051 32,36,52,27,40,48,36,28,58,24,60,30,48,64,48,20,66,48,44
%N A384051 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cubefull number.
%H A384051 Amiram Eldar, <a href="/A384051/b384051.txt">Table of n, a(n) for n = 1..10000</a>
%F A384051 Multiplicative with a(p^e) = p^e-1 if e <= 2, and p^e if e >= 3.
%F A384051 a(n) = n * A047994(n) / A384049(n).
%F A384051 a(n) = A047994(A360539(n)) * A360540(n).
%F A384051 Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(3*s-1)).
%F A384051 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.714093594477970831206... .
%t A384051 f[p_, e_] := p^e - If[e < 3, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384051 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] < 3, 1, 0));}
%Y A384051 Unitary analog of A384040.
%Y A384051 Cf. A036966, A360539, A360540, A384046, A384047.
%Y A384051 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), this sequence (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384051 nonn,easy,mult
%O A384051 1,3
%A A384051 _Amiram Eldar_, May 18 2025