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A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

%I A299149 #37 May 09 2025 05:54:26
%S A299149 1,1,3,3,5,3,7,5,27,5,11,9,13,7,15,35,17,27,19,15,21,11,23,15,75,13,
%T A299149 135,21,29,15,31,63,33,17,35,81,37,19,39,25,41,21,43,33,135,23,47,105,
%U A299149 147,75,51,39,53,135,55,35,57,29,59,45,61,31,189,231,65,33
%N A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).
%C A299149 Dirichlet convolution of a(n)/A046644(n) with itself yields A000265. - _Antti Karttunen_, Aug 30 2018
%H A299149 Antti Karttunen, <a href="/A299149/b299149.txt">Table of n, a(n) for n = 1..65537</a> (first 1000 terms Andrew Howroyd)
%H A299149 Vaclav Kotesovec, <a href="/A299149/a299149.jpg">Graph - the asymptotic ratio (65537 terms)</a>
%H A299149 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>.
%F A299149 a(n) = numerator(n*A317848(n)/A165825(n)) = A000265(n*A317848(n)). - _Andrew Howroyd_, Aug 09 2018
%F A299149 Sum_{k=1..n} A299149(k)/A299150(k) ~ n^2 / (2*sqrt(Pi*log(n))) * (1 + (1-gamma) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 09 2025
%e A299149 Sequence begins: 1, 1, 3/2, 3/2, 5/2, 3/2, 7/2, 5/2, 27/8, 5/2, 11/2, 9/4, 13/2, 7/2.
%t A299149 nn=50;
%t A299149 sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
%t A299149 Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
%t A299149 odd[n_] := n/2^IntegerExponent[n, 2]; f[p_, e_] := odd[p^e*Binomial[2*e, e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 30 2023 *)
%o A299149 (PARI) a(n)={my(v=factor(n)[,2]); numerator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ _Andrew Howroyd_, Aug 09 2018
%o A299149 (PARI) \\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
%o A299149 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}
%o A299149 apply(numerator, DirSqrt(vector(100, n, n))) \\ _Andrew Howroyd_, Aug 09 2018
%o A299149 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A299149 Cf. A000010, A000265, A003958, A007431, A018804, A023900, A029935, A046643, A046644, A165825, A257098, A298971, A299119, A299150 (denominators), A299151, A317848, A318319, A318321, A318649.
%K A299149 nonn,frac,mult
%O A299149 1,3
%A A299149 _Gus Wiseman_, Feb 03 2018
%E A299149 Keyword:mult added by _Andrew Howroyd_, Aug 09 2018