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A378547 a(n) is the sum of the divisors d of n for which A083345(n/d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

%I A378547 #9 Nov 30 2024 08:50:18
%S A378547 0,1,1,3,1,6,1,7,3,8,1,15,1,10,8,14,1,19,1,21,10,14,1,34,5,16,10,27,1,
%T A378547 40,1,29,14,20,12,48,1,22,16,48,1,52,1,39,25,26,1,69,7,41,20,45,1,60,
%U A378547 16,62,22,32,1,96,1,34,31,59,18,76,1,57,26,72,1,109,1,40,41,63,18,88,1,97,30,44,1,126,22,46,32
%N A378547 a(n) is the sum of the divisors d of n for which A083345(n/d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).
%C A378547 Dirichlet convolution of A000027 with A377874.
%C A378547 Dirichlet convolution of A000010 (Euler phi) with A378445.
%H A378547 Antti Karttunen, <a href="/A378547/b378547.txt">Table of n, a(n) for n = 1..20000</a>
%F A378547 a(n) = Sum_{d|n} A377874(n/d)*d.
%F A378547 a(n) = Sum_{d|n} A000010(n/d)*A378445(d).
%F A378547 a(n) = A000203(n) - A378546(n).
%o A378547 (PARI)
%o A378547 A377874(n) = { my(f=factor(n)); (numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1])))%2); };
%o A378547 A378547(n) = sumdiv(n,d,d*A377874(n/d));
%Y A378547 Cf. A000010, A000027, A000203, A083345, A377874, A378445, A378546.
%Y A378547 Cf. also A378544, A378545.
%K A378547 nonn
%O A378547 1,4
%A A378547 _Antti Karttunen_, Nov 30 2024