cp's OEIS Frontend

A317835 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A003415 (arithmetic derivative of n) + A063524 (1, 0, 0, 0, ...).

%I A317835 #6 Aug 26 2018 12:25:26
%S A317835 1,1,1,15,1,9,1,81,23,13,1,95,1,17,15,1499,1,127,1,151,19,25,1,393,39,
%T A317835 29,193,207,1,87,1,6311,27,37,23,969,1,41,31,661,1,119,1,319,259,49,1,
%U A317835 5499,55,295,39,375,1,769,31,929,43,61,1,593,1,65,347,50075,35,183,1,487,51,183,1,2751,1,77,371,543,35,215,1,9643,5611,85,1
%N A317835 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A003415 (arithmetic derivative of n) + A063524 (1, 0, 0, 0, ...).
%C A317835 The first negative term is a(240) = -5067.
%H A317835 Antti Karttunen, <a href="/A317835/b317835.txt">Table of n, a(n) for n = 1..65537</a>
%F A317835 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003415(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317835 (PARI)
%o A317835 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A317835 A317835aux(n) = if(1==n,n,(A003415(n)-sumdiv(n,d,if((d>1)&&(d<n),A317835aux(d)*A317835aux(n/d),0)))/2);
%o A317835 A317835(n) = numerator(A317835aux(n));
%Y A317835 Cf. A003415, A063524, A046644 (denominators).
%Y A317835 Cf. also A300251, A300252, A305809.
%K A317835 sign,frac
%O A317835 1,4
%A A317835 _Antti Karttunen_, Aug 12 2018