OFFSET
1,3
COMMENTS
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms Andrew Howroyd)
Wikipedia, Dirichlet convolution.
FORMULA
EXAMPLE
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.
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 *)
PROG
(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&&d<n, u[d]*u[n/d], 0)))/2); u}
apply(numerator, DirSqrt(vector(100, n, n))) \\ Andrew Howroyd, Aug 09 2018
CROSSREFS
KEYWORD
nonn,frac,mult
AUTHOR
Gus Wiseman, Feb 03 2018
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Aug 09 2018
STATUS
approved