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A384250 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is a powerful number.

%I A384250 #12 May 24 2025 00:22:04
%S A384250 1,1,2,4,4,2,6,6,9,4,10,8,12,6,8,16,16,9,18,16,12,10,22,12,25,12,21,
%T A384250 24,28,8,30,18,20,16,24,36,36,18,24,24,40,12,42,40,36,22,46,32,49,25,
%U A384250 32,48,52,21,40,36,36,28,58,32,60,30,54,64,48,20,66,64,44
%N A384250 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is a powerful number.
%H A384250 Amiram Eldar, <a href="/A384250/b384250.txt">Table of n, a(n) for n = 1..10000</a>
%F A384250 Multiplicative with a(p^e) = p^e if e is even, a(p) = p-1, and a(p^e) = p^e * (1 - 1/p + 1/p^A006519(e-1)) if e is odd >= 3.
%F A384250 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.7218055778498707651047..., and f(x) = (1 + x - x^2)/(1 + x) + (1 - x) * Sum_{k>=2} x^(2^k + 1) / (1 - x^(2^k));
%t A384250 f[p_, e_] := If[EvenQ[e], p^e, p^e*(1 - 1/p + 1/p^(2^IntegerExponent[e-1, 2]))]; f[p_, 1] := p-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384250 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, f[i,1]^f[i,2] * if(!(f[i,2]%2), 1, 1 - 1/f[i,1] + 1/f[i,1]^(1 << valuation(f[i,2]-1, 2)))));}
%Y A384250 Cf. A001694, A006519, A077609.
%Y A384250 Analogous sequences: A384039, A384050.
%Y A384250 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is: A384247(1), A384249 (squarefree), this sequence (powerful), A384251 (odd), A384252 (power of 2).
%K A384250 nonn,easy,mult
%O A384250 1,3
%A A384250 _Amiram Eldar_, May 23 2025