cp's OEIS Frontend

A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

%I A317933 #19 May 10 2025 04:11:41
%S A317933 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,
%T A317933 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,
%U A317933 1,1,1,1,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1
%N A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).
%C A317933 Multiplicative because A034444 is.
%C A317933 The first 2^20 terms are positive. Is the sequence nonnegative?
%C A317933 Records seem to be A001790, occurring at A000302 (apart from 4).
%H A317933 Antti Karttunen, <a href="/A317933/b317933.txt">Table of n, a(n) for n = 1..65537</a>
%H A317933 Vaclav Kotesovec, <a href="/A317933/a317933.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A317933 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A034444(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A317933 Sum_{k=1..n} A317933(k) / A317934(k) ~ sqrt(6)*n/Pi. - _Vaclav Kotesovec_, May 10 2025
%o A317933 (PARI)
%o A317933 A034444(n) = (2^omega(n));
%o A317933 A317933perA317934(n) = if(1==n,n,(A034444(n)-sumdiv(n,d,if((d>1)&&(d<n),A317933perA317934(d)*A317933perA317934(n/d),0)))/2);
%o A317933 A317933(n) = numerator(A317933perA317934(n));
%o A317933 (PARI)
%o A317933 up_to = 65537;
%o A317933 \\ Faster:
%o A317933 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 A317933 v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
%o A317933 A317933(n) = numerator(v317933aux[n]);
%o A317933 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A317933 Cf. A001790, A034444, A317934 (denominators).
%Y A317933 Cf. also A046643, A317831, A317925, A317937.
%K A317933 nonn,frac,mult
%O A317933 1,16
%A A317933 _Antti Karttunen_, Aug 12 2018