A048243 Number of non-unitary divisors of binomial(n, floor(n/2)).

0, 0, 0, 0, 0, 2, 0, 0, 4, 10, 0, 8, 8, 16, 8, 16, 0, 16, 0, 16, 32, 64, 0, 32, 80, 128, 224, 320, 256, 352, 64, 128, 256, 512, 256, 640, 640, 1024, 512, 1280, 512, 1024, 512, 1024, 2560, 3328, 1280, 2432, 2432, 3584, 3584, 5120, 2048, 2816, 2048, 4096, 8192

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Examples

			For n = 10, binomial(10,5) = 252 = 4*9*7 has 18 divisors, 8 are unitary and the residual 10 are non-unitary; thus a(10) = 18 - 8 = 10.
		

Crossrefs

Cf. A000005, A001405, A034444, A048105.

Programs

Mathematica

Table[Function[k, DivisorSum[k, 1 &, ! CoprimeQ[#, k/#] &]]@ Binomial[n, Ceiling[n/2]], {n, 57}] (* Michael De Vlieger, Jun 29 2017 *)
f[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Table[f[Binomial[n, Floor[n/2]]], {n, 1, 60}] (* Amiram Eldar, Jul 22 2024 *)

PARI

a(n) = apply(x -> numdiv(x) - 2^omega(x), binomial(n, n\2)); \\ Amiram Eldar, Jul 22 2024

Formula

a(n) = A048105(A001405(n)).

a(n) = A000005(A001405(n)) - A034444(A001405(n)). [corrected by Amiram Eldar, Jul 22 2024]