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A378546 a(n) is the sum of the divisors d of n for which A083345(n/d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

%I A378546 #7 Nov 30 2024 08:50:15
%S A378546 1,2,3,4,5,6,7,8,10,10,11,13,13,14,16,17,17,20,19,21,22,22,23,26,26,
%T A378546 26,30,29,29,32,31,34,34,34,36,43,37,38,40,42,41,44,43,45,53,46,47,55,
%U A378546 50,52,52,53,53,60,56,58,58,58,59,72,61,62,73,68,66,68,67,69,70,72,71,86,73,74,83,77,78,80,79,89,91
%N A378546 a(n) is the sum of the divisors d of n for which A083345(n/d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).
%C A378546 Dirichlet convolution of A000027 with A369001.
%C A378546 Dirichlet convolution of A000010 (Euler phi) with A378444.
%H A378546 Antti Karttunen, <a href="/A378546/b378546.txt">Table of n, a(n) for n = 1..20000</a>
%F A378546 a(n) = Sum_{d|n} A369001(n/d)*d.
%F A378546 a(n) = Sum_{d|n} A000010(n/d)*A378444(d).
%F A378546 a(n) = A000203(n) - A378547(n).
%o A378546 (PARI)
%o A378546 A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
%o A378546 A369001(n) = !(A083345(n)%2);
%o A378546 A378546(n) = sumdiv(n,d,d*A369001(n/d));
%Y A378546 Cf. A000010, A000027, A000203, A083345, A369001, A378444, A378547.
%Y A378546 Cf. also A378544, A378545.
%K A378546 nonn
%O A378546 1,2
%A A378546 _Antti Karttunen_, Nov 30 2024