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A299149
Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).
9
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
1,3
COMMENTS
Dirichlet convolution of a(n)/A046644(n) with itself yields A000265. - Antti Karttunen, Aug 30 2018
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms Andrew Howroyd)
FORMULA
a(n) = numerator(n*A317848(n)/A165825(n)) = A000265(n*A317848(n)). - Andrew Howroyd, Aug 09 2018
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
KEYWORD
nonn,frac,mult
AUTHOR
Gus Wiseman, Feb 03 2018
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Aug 09 2018
STATUS
approved