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A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).

%I A317846 #5 Aug 23 2018 21:03:01
%S A317846 1,1,1,7,1,7,1,25,11,11,1,43,1,15,13,363,1,71,1,67,17,23,1,139,19,27,
%T A317846 61,91,1,57,1,1335,25,35,21,365,1,39,29,215,1,81,1,139,131,47,1,1875,
%U A317846 27,199,37,163,1,367,29,291,41,59,1,235,1,63,171,9923,33,129,1,211,49,137,1,1055,1,75,235,235,33,153,1,2883,1363,83,1,335,41
%N A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).
%C A317846 The first negative term is a(420) = -1269.
%H A317846 Antti Karttunen, <a href="/A317846/b317846.txt">Table of n, a(n) for n = 1..65537</a>
%F A317846 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A051953(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317846 (PARI)
%o A317846 A317846aux(n) = if(1==n,n,((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(d<n),A317846aux(d)*A317846aux(n/d),0)))/2);
%o A317846 A317846(n) = numerator(A317846aux(n));
%o A317846 (PARI)
%o A317846 \\ Memoized implementation:
%o A317846 memo317846 = Map();
%o A317846 A317846aux(n) = if(1==n,n,if(mapisdefined(memo317846,n),mapget(memo317846,n),my(v = ((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(d<n),A317846aux(d)*A317846aux(n/d),0)))/2); mapput(memo317846,n,v); (v)));
%Y A317846 Cf. A051953, A063524, A046644 (denominators).
%Y A317846 Cf. also A317845, A317925.
%K A317846 sign,frac
%O A317846 1,4
%A A317846 _Antti Karttunen_, Aug 12 2018