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

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, 135, 21, 29, 15, 31, 63, 33, 17, 35, 81, 37, 19, 39, 25, 41, 21, 43, 33, 135, 23, 47, 105, 147, 75, 51, 39, 53, 135, 55, 35, 57, 29, 59, 45, 61, 31, 189, 231, 65, 33

Offset: 1

Author

Gus Wiseman, Feb 03 2018

Keywords

• nonn
• frac
• mult

Comments

Dirichlet convolution of a(n)/A046644(n) with itself yields A000265. - Antti Karttunen, Aug 30 2018

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms Andrew Howroyd)
• Vaclav Kotesovec, Graph - the asymptotic ratio (65537 terms)
• Wikipedia, Dirichlet convolution.

Crossrefs

Cf. A000010, A000265, A003958, A007431, A018804, A023900, A029935, A046643, A046644, A165825, A257098, A298971, A299119, A299150 (denominators), A299151, A317848, A318319, A318321, A318649.

Programs

Mathematica

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
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 *)

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

PARI

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
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&&dAndrew Howroyd, Aug 09 2018

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Formula

a(n) = numerator(n*A317848(n)/A165825(n)) = A000265(n*A317848(n)). - Andrew Howroyd, Aug 09 2018

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

Extensions

Keyword:mult added by Andrew Howroyd, Aug 09 2018