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A384053 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 cube.

%I A384053 #8 May 21 2025 01:33:24
%S A384053 1,1,2,3,4,2,6,8,8,4,10,6,12,6,8,15,16,8,18,12,12,10,22,16,24,12,27,
%T A384053 18,28,8,30,31,20,16,24,24,36,18,24,32,40,12,42,30,32,22,46,30,48,24,
%U A384053 32,36,52,27,40,48,36,28,58,24,60,30,48,64,48,20,66,48,44
%N A384053 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 cube.
%H A384053 Amiram Eldar, <a href="/A384053/b384053.txt">Table of n, a(n) for n = 1..10000</a>
%F A384053 Multiplicative with a(p^e) = p^e if e is a multiple of 3, and p^e-1 otherwise.
%F A384053 Dirichlet g.f.: zeta(s-1) * zeta(3*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) - 1/p^(3*s) + 1/p^(3*s-1)).
%F A384053 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.71190515701599590826... .
%t A384053 f[p_, e_] := p^e - If[Divisible[e, 3], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384053 (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 A384053 Unitary analog of A078429.
%Y A384053 Cf. A013664, A384046, A384047.
%Y A384053 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), A384051 (cubefull), A384052 (square), this sequence (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384053 nonn,easy,mult
%O A384053 1,3
%A A384053 _Amiram Eldar_, May 18 2025