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A384050 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 powerful number.

%I A384050 #10 May 21 2025 01:32:21
%S A384050 1,1,2,4,4,2,6,8,9,4,10,8,12,6,8,16,16,9,18,16,12,10,22,16,25,12,27,
%T A384050 24,28,8,30,32,20,16,24,36,36,18,24,32,40,12,42,40,36,22,46,32,49,25,
%U A384050 32,48,52,27,40,48,36,28,58,32,60,30,54,64,48,20,66,64,44
%N A384050 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 powerful number.
%H A384050 Amiram Eldar, <a href="/A384050/b384050.txt">Table of n, a(n) for n = 1..10000</a>
%F A384050 Multiplicative with a(p) = p-1, and p^e if e >= 2.
%F A384050 a(n) = n * A047994(n) / A384048(n).
%F A384050 a(n) = A047994(A055231(n)) * A057521(n) = A000010(A055231(n)) * A057521(n).
%F A384050 Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1)).
%F A384050 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
%t A384050 f[p_, e_] := p^e - If[e < 2, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384050 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] == 1, 1, 0));}
%Y A384050 Unitary analog of A384039.
%Y A384050 Cf. A000010, A001694, A055231, A057521, A330596, A384046, A384047.
%Y A384050 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), this sequence (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384050 nonn,easy,mult
%O A384050 1,3
%A A384050 _Amiram Eldar_, May 18 2025