A056673 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).

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, 32, 32, 64, 128, 128, 64, 256, 128, 128, 128, 512, 256, 512, 512, 512, 512, 64, 64, 256, 128, 128, 128, 128, 128, 256, 256, 2048, 2048, 4096, 4096, 2048, 2048, 2048, 2048

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000005(A055231(x)) = A000005(A007913(x)/A055229(x)), where x = A001405(n) = binomial(n, floor(n/2)).

a(n) = A056671(A001405(n)). - Amiram Eldar, Sep 06 2020

Example

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.

Mathematica

Table[With[{m = Binomial[n, Floor[n/2]]}, DivisorSum[m, 1 &, And[CoprimeQ[#, m/#], SquareFreeQ@ #] &]], {n, 62}] (* Michael De Vlieger, Sep 05 2017 *)
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 *)

Prog

(PARI) a(n) = my(b=binomial(n, n\2)); sumdiv(b, d, issquarefree(d) && (gcd(d, b/d) == 1)); \\ Michel Marcus, Sep 05 2017

Crossrefs

Cf. A000005, A001405, A007913, A055229, A055231, A056060, A056671.

Sequence in context: A352502 A238004 A048244 * A353761 A128442 A290633

Adjacent sequences: A056670 A056671 A056672 * A056674 A056675 A056676

Keyword

nonn

Author

Labos Elemer, Aug 10 2000

Status

approved