A056649 - OEIS

%I #15 Sep 28 2024 07:37:48

%S 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,

%T 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,

%U 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

%N a(n) = A056061(n) - A034444(A056647(n)).

%C 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

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

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

%t 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]];

%t 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 *)

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

%K nonn

%O 1,26

%A _Labos Elemer_, Aug 09 2000

%E Incorrect name replaced with a formula by _Amiram Eldar_, Sep 28 2024