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A384052 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 square.

%I A384052 #8 May 21 2025 01:32:27
%S A384052 1,1,2,4,4,2,6,7,9,4,10,8,12,6,8,16,16,9,18,16,12,10,22,14,25,12,26,
%T A384052 24,28,8,30,31,20,16,24,36,36,18,24,28,40,12,42,40,36,22,46,32,49,25,
%U A384052 32,48,52,26,40,42,36,28,58,32,60,30,54,64,48,20,66,64,44
%N A384052 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 square.
%H A384052 Amiram Eldar, <a href="/A384052/b384052.txt">Table of n, a(n) for n = 1..10000</a>
%F A384052 Multiplicative with a(p^e) = p^e-1 if e is odd, and p^e if e is even.
%F A384052 a(n) = n * A047994(n) / A384054(n).
%F A384052 a(n) = A047994(A350389(n)) * A350388(n).
%F A384052 Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1)).
%F A384052 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.74061963657217328604... .
%t A384052 f[p_, e_] := p^e - If[OddQ[e], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384052 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%2, 1, 0));}
%Y A384052 Unitary analog of A206369.
%Y A384052 Cf. A013662, A350388, A350389, A384046, A384047.
%Y A384052 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), this sequence (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384052 nonn,easy,mult
%O A384052 1,3
%A A384052 _Amiram Eldar_, May 18 2025