cp's OEIS Frontend

A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

%I A317941 #7 Aug 24 2018 22:12:38
%S A317941 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 A317941 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 A317941 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 A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.
%C A317941 Multiplicative because A037445 is.
%H A317941 Antti Karttunen, <a href="/A317941/b317941.txt">Table of n, a(n) for n = 1..65537</a>
%F A317941 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 A317941 (PARI)
%o A317941 up_to = 1+(2^16);
%o A317941 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 A317941 A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ From A037445
%o A317941 v317941aux = DirSqrt(vector(up_to, n, A037445(n)));
%o A317941 A317941(n) = numerator(v317941aux[n]);
%Y A317941 Cf. A037445, A317934 (denominators).
%Y A317941 Cf. also A317933, A317940.
%K A317941 sign,frac,mult
%O A317941 1,8
%A A317941 _Antti Karttunen_, Aug 22 2018