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Number of non-unitary divisors of binomial(n, floor(n/2)).
4

%I #18 Jul 22 2024 15:22:34

%S 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,

%T 320,256,352,64,128,256,512,256,640,640,1024,512,1280,512,1024,512,

%U 1024,2560,3328,1280,2432,2432,3584,3584,5120,2048,2816,2048,4096,8192

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

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

%F a(n) = A000005(A001405(n)) - A034444(A001405(n)). [corrected by _Amiram Eldar_, Jul 22 2024]

%e 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.

%t Table[Function[k, DivisorSum[k, 1 &, ! CoprimeQ[#, k/#] &]]@ Binomial[n, Ceiling[n/2]], {n, 57}] (* _Michael De Vlieger_, Jun 29 2017 *)

%t f[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Table[f[Binomial[n, Floor[n/2]]], {n, 1, 60}] (* _Amiram Eldar_, Jul 22 2024 *)

%o (PARI) a(n) = apply(x -> numdiv(x) - 2^omega(x), binomial(n, n\2)); \\ _Amiram Eldar_, Jul 22 2024

%Y Cf. A000005, A001405, A034444, A048105.

%K nonn

%O 1,6

%A _Labos Elemer_