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A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.

%I A317927 #7 Aug 11 2018 22:00:57
%S A317927 1,3,2,19,4,2,11,63,6,3,19,13,23,17,5,867,16,4,35,5,17,25,21,11,31,29,
%T A317927 13,113,27,13,57,3069,13,9,23,25,71,41,14,69,79,33,41,169,9,25,89,615,
%U A317927 259,53,17,197,51,25,29,389,20,31,113,59,117,67,10,22199,18,14,131,31,51,71,69,11,143,77,22,281,91,35,153,489,71,85,81,151,19
%N A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.
%C A317927 The first negative term is a(330) = -21.
%H A317927 Antti Karttunen, <a href="/A317927/b317927.txt">Table of n, a(n) for n = 1..16384</a>
%F A317927 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A005187(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317927 (PARI)
%o A317927 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A317927 A317927perA317928(n) = if(1==n,n,(A005187(n)-sumdiv(n,d,if((d>1)&&(d<n),A317927perA317928(d)*A317927perA317928(n/d),0)))/2);
%o A317927 A317927(n) = numerator(A317927perA317928(n));
%o A317927 (PARI)
%o A317927 \\ Memoized implementation:
%o A317927 memo = Map();
%o A317927 A317927perA317928(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A005187(n)-sumdiv(n,d,if((d>1)&&(d<n),A317927perA317928(d)*A317927perA317928(n/d),0)))/2); mapput(memo,n,v); (v)));
%Y A317927 Cf. A005187, A317928 (denominators).
%Y A317927 Cf. also A297111, A300244, A299151, A317931.
%K A317927 sign,frac
%O A317927 1,2
%A A317927 _Antti Karttunen_, Aug 11 2018