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A048784

a(n) = tau(binomial(2*n,n)), where tau = number of divisors (A000005).

1, 2, 4, 6, 8, 18, 24, 32, 48, 48, 48, 128, 96, 192, 384, 480, 384, 768, 1152, 1536, 2304, 2048, 2048, 3840, 3456, 4608, 6144, 3840, 8192, 20480, 10240, 12288, 18432, 36864, 36864, 49152, 24576, 32768, 98304, 92160, 73728, 245760, 262144

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

G. V. Fedorov, Number of divisors of the central binomial coefficient, Moscow Univ. Math. Bull., Vol. 68 (2013), pp. 194-197.

FORMULA

a(n) = A000005(A000984(n)). - Michel Marcus, Aug 21 2013

log(a(n)) = log(2) * (pi(2*n)-pi(n)) + log(2) * (n/log(n)) * Sum_{k=0..T} c_k/log(n)^k + O(n/log(n)^(T+2)) for any T >= 0, where c_k = Sum_{m>=1} Integral_{m+1/2..m+1} log(t)^m/t^2 dt. In particular for T = 0, log(a(n)) = 2 * log(2)^2 * (n/log(n)) + O(n/log(n)^2) (Fedorov, 2013). - Amiram Eldar, Dec 10 2024

MAPLE

A048784 := proc(n)

numtheory[tau](binomial(2*n, n)) ;

end proc:

seq(A048784(n), n=0..30) ; # R. J. Mathar, Jul 12 2024

MATHEMATICA

f[n_] := DivisorSigma[0, Binomial[2 n, n]]; Table[f@n, {n, 0, 42}] (* Robert G. Wilson v, Apr 08 2009 *)

PROG

(PARI) fv(n, p)=my(s); while(n\=p, s+=n); s

a(n)=my(s=1); forprime(p=2, 2*n, s*=fv(2*n, p)-2*fv(n, p)+1); s \\ Charles R Greathouse IV, Aug 21 2013

CROSSREFS

Cf. A000005, A000984, A002162.