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%I #7 Aug 24 2018 22:13:02
%S 1,2,1,8,1,2,1,16,2,1,1,8,1,2,1,128,1,4,1,4,1,2,1,16,1,1,2,8,1,1,1,
%T 256,1,1,1,16,1,2,1,8,1,2,1,8,1,2,1,128,2,1,1,4,1,4,1,16,1,1,1,4,1,2,
%U 2,1024,1,2,1,1,1,1,1,32,1,1,1,8,1,1,1,64,8,1,1,8,1,2,1,16,1,2,1,8,1,2,1,256,1,4,2,1,1,1,1,8,1
%N Denominators of rational valued sequence whose Dirichlet convolution with itself yields A173557.
%C Not multiplicative because A318317 contains zeros.
%C Differs from A317926 at n = 200, 400, 600, 800, 900, 1200, 1400, 1600, 1800, 2200, 2400, 2700, 2800, 3200, 3600, 3800, 4050, 4200, 4400, 4600, 4800, 4900, 5200, ..., which seem to be a subsequence of positions of zeros in A318317.
%C Here a(200) = 1, while A317926(200) = 2.
%H Antti Karttunen, <a href="/A318318/b318318.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A173557(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o (PARI)
%o up_to = 16384;
%o A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
%o DirSqrt(v) = {my(n=#v, u=vector(nA173557)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
%o v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
%o A318318(n) = denominator(v318317_18[n]);
%Y Cf. A173557, A318317 (numerators).
%Y Cf. also A317926.
%K nonn,frac
%O 1,2
%A _Antti Karttunen_, Aug 24 2018
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