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A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.

%I A384048 #11 May 21 2025 01:32:16
%S A384048 1,2,3,3,5,6,7,7,8,10,11,9,13,14,15,15,17,16,19,15,21,22,23,21,24,26,
%T A384048 26,21,29,30,31,31,33,34,35,24,37,38,39,35,41,42,43,33,40,46,47,45,48,
%U A384048 48,51,39,53,52,55,49,57,58,59,45,61,62,56,63,65,66,67,51
%N A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.
%H A384048 Amiram Eldar, <a href="/A384048/b384048.txt">Table of n, a(n) for n = 1..10000</a>
%F A384048 Multiplicative with a(p) = p and a(p^e) = p^e - 1 if e >= 2.
%F A384048 a(n) = n * A047994(n) / A384050(n).
%F A384048 a(n) = A047994(A057521(n)) * A055231(n) = A000010(A055231(n)) * A057521(n).
%F A384048 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(3*s-1)).
%F A384048 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^3*(p+1))) = 0.947733... (A065466).
%t A384048 f[p_, e_] := If[e == 1, p, p^e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384048 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] == 1, 0, 1));}
%Y A384048 Unitary analog of A063659.
%Y A384048 Cf. A000010, A005117, A055231, A057521, A065466, A384046, A384047.
%Y A384048 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), this sequence (squarefree), A384049 (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 A384048 nonn,easy,mult
%O A384048 1,2
%A A384048 _Amiram Eldar_, May 18 2025