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A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.

%I A384049 #10 May 21 2025 01:32:09
%S A384049 1,2,3,4,5,6,7,7,9,10,11,12,13,14,15,15,17,18,19,20,21,22,23,21,25,26,
%T A384049 26,28,29,30,31,31,33,34,35,36,37,38,39,35,41,42,43,44,45,46,47,45,49,
%U A384049 50,51,52,53,52,55,49,57,58,59,60,61,62,63,63,65,66,67,68
%N A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.
%H A384049 Amiram Eldar, <a href="/A384049/b384049.txt">Table of n, a(n) for n = 1..10000</a>
%F A384049 Multiplicative with a(p^e) = p^e if e <= 2, and p^e - 1 if e >= 3.
%F A384049 a(n) = n * A047994(n) / A384051(n).
%F A384049 a(n) = A047994(A360540(n)) * A360539(n).
%F A384049 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(3*s) + 1/p^(4*s-1)).
%F A384049 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^5*(p+1))) = 0.988504... (A065468).
%t A384049 f[p_, e_] := p^e - If[e < 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384049 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] < 3, 0, 1));}
%Y A384049 Unitary analog of A254926.
%Y A384049 Cf. A004709, A065468, A360539, A360540, A384046, A384047.
%Y A384049 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), this sequence (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384049 nonn,easy,mult
%O A384049 1,2
%A A384049 _Amiram Eldar_, May 18 2025