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A384056 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 power of 2.

%I A384056 #8 May 21 2025 01:33:07
%S A384056 1,2,2,4,4,4,6,8,8,8,10,8,12,12,8,16,16,16,18,16,12,20,22,16,24,24,26,
%T A384056 24,28,16,30,32,20,32,24,32,36,36,24,32,40,24,42,40,32,44,46,32,48,48,
%U A384056 32,48,52,52,40,48,36,56,58,32,60,60,48,64,48,40,66,64,44
%N A384056 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 power of 2.
%H A384056 Amiram Eldar, <a href="/A384056/b384056.txt">Table of n, a(n) for n = 1..10000</a>
%F A384056 Multiplicative with a(2^e) = 2^e, and p^e-1 if p is an odd prime.
%F A384056 a(n) = n * A047994(n) / A384055(n).
%F A384056 a(n) = A047994(A000265(n)) * A006519(n).
%F A384056 Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1-1/2^s)/(1-1/2^(s-1)+1/2^(2*s-1))) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)).
%F A384056 Sum_{k=1..n} a(k) ~ (3/5) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.
%t A384056 f[p_, e_] := p^e - If[p == 2, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384056 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] == 2, 0, 1));}
%Y A384056 Unitary analog of A062570.
%Y A384056 Cf. A000079, A000265, A006519, A065463, A384046, A384047.
%Y A384056 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), A384055 (odd), this sequence (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384056 nonn,easy,mult
%O A384056 1,2
%A A384056 _Amiram Eldar_, May 18 2025