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A384055 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is odd.

%I A384055 #8 May 21 2025 01:33:18
%S A384055 1,1,3,3,5,3,7,7,9,5,11,9,13,7,15,15,17,9,19,15,21,11,23,21,25,13,27,
%T A384055 21,29,15,31,31,33,17,35,27,37,19,39,35,41,21,43,33,45,23,47,45,49,25,
%U A384055 51,39,53,27,55,49,57,29,59,45,61,31,63,63,65,33,67,51,69
%N A384055 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is odd.
%H A384055 Amiram Eldar, <a href="/A384055/b384055.txt">Table of n, a(n) for n = 1..10000</a>
%F A384055 Multiplicative with a(2^e) = 2^e-1, and p^e if p is an odd prime.
%F A384055 a(n) = n * A047994(n) / A384056(n).
%F A384055 a(n) = A047994(A006519(n)) * A000265(n).
%F A384055 Dirichlet g.f.: zeta(s-1) * (1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s).
%F A384055 Sum_{k=1..n} a(k) ~ (5/12) * n^2.
%t A384055 f[p_, e_] := p^e - If[p == 2, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384055 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] == 2, 1, 0));}
%Y A384055 Unitary analog of A026741.
%Y A384055 Cf. A000265, A006519, A384046, A384047.
%Y A384055 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), A384053 (cube), A384054 (exponentially odd), this sequence (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384055 nonn,easy,mult
%O A384055 1,3
%A A384055 _Amiram Eldar_, May 18 2025