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A384054 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is an exponentially odd number.

%I A384054 #12 May 21 2025 23:19:01
%S A384054 1,2,3,3,5,6,7,8,8,10,11,9,13,14,15,15,17,16,19,15,21,22,23,24,24,26,
%T A384054 27,21,29,30,31,32,33,34,35,24,37,38,39,40,41,42,43,33,40,46,47,45,48,
%U A384054 48,51,39,53,54,55,56,57,58,59,45,61,62,56,63,65,66,67,51
%N A384054 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is an exponentially odd number.
%H A384054 Amiram Eldar, <a href="/A384054/b384054.txt">Table of n, a(n) for n = 1..10000</a>
%F A384054 Multiplicative with a(p^e) = p^e if e is odd, and p^e-1 if e is even.
%F A384054 a(n) = n * A047994(n) / A384052(n).
%F A384054 a(n) = A047994(A350388(n)) * A350389(n).
%F A384054 Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s-1)).
%F A384054 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.95692470821076622881... .
%t A384054 f[p_, e_] := p^e - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
%o A384054 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%2, 0, 1));}
%o A384054 (Python)
%o A384054 from math import prod
%o A384054 from sympy import factorint
%o A384054 def A384054(n): return prod(p**e-(e&1^1) for p,e in factorint(n).items()) # _Chai Wah Wu_, May 21 2025
%Y A384054 Unitary analog of A384041.
%Y A384054 Cf. A013662, A268335, A350388, A350389, A384046, A384047.
%Y A384054 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), this sequence (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).
%K A384054 nonn,easy,mult
%O A384054 1,2
%A A384054 _Amiram Eldar_, May 18 2025