A056649 a(n) = A056061(n) - A034444(A056647(n)).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 4, 6, 2, 2, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 6, 8, 0, 0, 0, 4, 4, 6, 2, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 4, 4, 0, 0, 4, 8, 2, 3, 6, 8, 4, 8, 2, 2, 4, 4, 8, 8, 0, 0, 0, 4, 2, 4, 3, 4, 2, 3, 4

Offset

1,26

Comments

Previous name, "Number of non-unitary square divisors of central binomial coefficient", was incorrect. See A376556 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

Links

Table of n, a(n) for n=1..105.

Formula

a(n) = A056061(n) - 2^r, where r = A001221(A000188(A001405(n))/A055229(A001405(n))).

Example

a(28) = A056061(28) - A034444(A056647(28)) = A056061(28) - A034444(25) = 8 - 2 = 6.

Mathematica

A056061[n_] := Count[Divisors@Binomial[n, Floor[n/2]], d_ /; IntegerQ@Sqrt@d]; A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]];
Table[A056061[n] - 2^(PrimeNu[Sqrt[A008833[Binomial[n, Floor[n/2]]]]/ A055229[Binomial[n, Floor[n/2]]]]), {n, 1, 15}] (* G. C. Greubel, May 20 2017 *)

Crossrefs

Cf. A000188, A001221, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061, A376556.

Sequence in context: A254350 A357049 A364864 * A376556 A075242 A161489

Adjacent sequences: A056646 A056647 A056648 * A056650 A056651 A056652

Keyword

nonn

Author

Labos Elemer, Aug 09 2000

Extensions

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

Status

approved