%I #23 Sep 06 2020 03:46:19
%S 1,2,2,4,4,2,4,8,4,2,16,8,8,8,8,16,32,16,32,16,32,32,64,32,16,16,8,8,
%T 32,32,64,128,128,64,256,128,128,128,512,256,512,512,512,512,64,64,
%U 256,128,128,128,128,128,256,256,2048,2048,4096,4096,2048,2048,2048,2048
%N Number of unitary and squarefree divisors of binomial(n, floor(n/2)). Also the number of divisors of the powerfree part of A001405(n), A056060(n).
%H Amiram Eldar, <a href="/A056673/b056673.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000005(A055231(x)) = A000005(A007913(x)/A055229(x)), where x = A001405(n) = binomial(n, floor(n/2)).
%F a(n) = A056671(A001405(n)). - _Amiram Eldar_, Sep 06 2020
%e n = 14: binomial(15,7) = 3432 = 2*2*2*3*11*13, which has 32 divisors. Of those divisors, 16 are unitary: {1, 3, 8, 11, 13, 24, 33, 39, 88, 104, 143, 264, 312, 429, 1144, 3432}; 16 are squarefree: {1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858}. Only 8 of the divisors belong to both classes: {1, 3, 11, 13, 33, 39, 143, 429}. Thus, a(14) = 8.
%t Table[With[{m = Binomial[n, Floor[n/2]]}, DivisorSum[m, 1 &, And[CoprimeQ[#, m/#], SquareFreeQ@ #] &]], {n, 62}] (* _Michael De Vlieger_, Sep 05 2017 *)
%t f[p_, e_] := If[e == 1, 2, 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[ Binomial[n, Floor[n/2]]]); Array[a, 60] (* _Amiram Eldar_, Sep 06 2020 *)
%o (PARI) a(n) = my(b=binomial(n, n\2)); sumdiv(b, d, issquarefree(d) && (gcd(d, b/d) == 1)); \\ _Michel Marcus_, Sep 05 2017
%Y Cf. A000005, A001405, A007913, A055229, A055231, A056060, A056671.
%K nonn
%O 1,2
%A _Labos Elemer_, Aug 10 2000