cp's OEIS Frontend

A318321 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003961.

%I A318321 #6 Aug 26 2018 12:27:41
%S A318321 1,3,5,27,7,15,11,135,75,21,13,135,17,33,35,2835,19,225,23,189,55,39,
%T A318321 29,675,147,51,625,297,31,105,37,15309,65,57,77,2025,41,69,85,945,43,
%U A318321 165,47,351,525,87,53,14175,363,441,95,459,59,1875,91,1485,115,93,61,945,67,111,825,168399,119,195,71,513,145,231,73
%N A318321 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003961.
%C A318321 Multiplicative because A003961 is.
%H A318321 Antti Karttunen, <a href="/A318321/b318321.txt">Table of n, a(n) for n = 1..16384</a>
%H A318321 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A318321 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003961(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A318321 (PARI)
%o A318321 up_to = 16384;
%o A318321 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A318321 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 A318321 v318321aux = DirSqrt(vector(up_to, n, A003961(n)));
%o A318321 A318321(n) = numerator(v318321aux[n]);
%Y A318321 Cf. A003961, A046644 (denominators).
%Y A318321 Cf. also A318319.
%K A318321 nonn,frac,mult
%O A318321 1,2
%A A318321 _Antti Karttunen_, Aug 24 2018