%I #7 Aug 24 2018 22:12:38

%S 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,-5,1,1,1,1,1,1,1,3,1,1,3,1,1,1,1,15,1,

%T 1,1,1,1,1,1,3,1,1,1,1,1,1,1,-5,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,-11,1,1,

%U 1,1,1,1,1,3,1,1,1,1,1,1,1,-5,-5,1,1,1,1,1,1,3,1,1,1,1,1,1,1,15,1,1,1,1,1,1,1,3,1

%N Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

%C Multiplicative because A037445 is.

%H Antti Karttunen, <a href="/A317941/b317941.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A037445(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

%o (PARI)

%o up_to = 1+(2^16);

%o DirSqrt(v) = {my(n=#v, u=vector(n)); 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 A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ From A037445

%o v317941aux = DirSqrt(vector(up_to, n, A037445(n)));

%o A317941(n) = numerator(v317941aux[n]);

%Y Cf. A037445, A317934 (denominators).

%Y Cf. also A317933, A317940.

%K sign,frac,mult

%O 1,8

%A _Antti Karttunen_, Aug 22 2018