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A378444 a(n) is the number of divisors d of n such that A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

%I A378444 #14 Jul 08 2025 07:52:11
%S A378444 1,1,1,1,1,1,1,1,2,1,1,2,1,1,2,2,1,2,1,2,2,1,1,2,2,1,2,2,1,2,1,2,2,1,
%T A378444 2,3,1,1,2,2,1,2,1,2,3,1,1,3,2,2,2,2,1,2,2,2,2,1,1,4,1,1,3,2,2,2,1,2,
%U A378444 2,2,1,3,1,1,3,2,2,2,1,3,3,1,1,4,2,1,2,2,1,3,2,2,2,1,2,3,1,2,3,3,1,2,1,2,4
%N A378444 a(n) is the number of divisors d of n such that A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).
%C A378444 Number of terms of A369002 that divide n.
%H A378444 Antti Karttunen, <a href="/A378444/b378444.txt">Table of n, a(n) for n = 1..65537</a>
%H A378444 Jon Maiga, <a href="http://sequencedb.net/s/A378444">Computer-generated formulas for A378444</a>, Sequence Machine.
%F A378444 a(n) = Sum_{d|n} A369001(d).
%F A378444 a(n) = A000005(n) - A378445(n).
%F A378444 a(n) = Sum_{d|n} A023900(d)*A378546(n/d).
%F A378444 a(n) = ceiling(A174273(n)/2). [Conjectured] - _Antti Karttunen_, May 14 2025
%o A378444 (PARI)
%o A378444 A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
%o A378444 A369001(n) = !(A083345(n)%2);
%o A378444 A378444(n) = sumdiv(n,d,A369001(d));
%Y A378444 Inverse Möbius transform of A369001.
%Y A378444 Cf. A000005, A174273, A369002, A378445, A378546.
%Y A378444 Cf. also A369257.
%K A378444 nonn
%O A378444 1,9
%A A378444 _Antti Karttunen_, Nov 27 2024