cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

This page as a plain text file.
%I A318453 #14 May 09 2025 10:47:52
%S A318453 1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,35,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,63,1,
%T A318453 1,1,3,1,1,1,5,1,1,1,3,1,1,1,35,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,231,1,1,
%U A318453 1,3,1,1,1,5,1,1,1,3,1,1,1,35,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,63,1,1,1,3,1,1,1,5,1
%N A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.
%H A318453 Antti Karttunen, <a href="/A318453/b318453.txt">Table of n, a(n) for n = 1..65537</a>
%H A318453 Vaclav Kotesovec, <a href="/A318453/a318453.jpg">Graph - the asymptotic ratio (16384 terms)</a>
%F A318453 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318453 Sum_{k=1..n} A318453(k) / A318454(k) ~ n/sqrt(2). - _Vaclav Kotesovec_, May 09 2025
%t A318453 f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
%t A318453 Table[f[n] // Numerator, {n, 1, 105}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318453 (PARI)
%o A318453 up_to = 16384;
%o A318453 A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
%o A318453 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 A318453 v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
%o A318453 A318453(n) = numerator(v318453_54[n]);
%o A318453 A318454(n) = denominator(v318453_54[n]);
%Y A318453 Cf. A001227.
%Y A318453 Cf. A318454 (gives the denominators).
%Y A318453 Differs from A318313 for the first time at n=81, where a(81) = 1, while A318313(81) = 3.
%K A318453 nonn,frac
%O A318453 1,4
%A A318453 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018