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%I #11 Sep 05 2017 03:09:06
%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 Number of non-unitary square divisors of central binomial coefficient.
%F a(n) = A056061(n) - 2^r, where r is the number of prime factors in the largest unitary square (or square root) divisor of central binomial coefficient: r=A001221(A000188(A001405(n))/A055229(A001405(n))).
%e n=28: binomial(28,14) = 2*2*2*3*3*3*5*5*17*19*23. It has 384 divisors of which 8 are also square numbers: {1, 4, 9, 25, 36, 100, 225, 900} and only {4, 9, 36, 100, 225, 900} are not unitary divisors. Thus a(28) = 8 - 2 = 6. Observe large values (e.g., 223), where a(n)=0.
%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, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061.
%K nonn
%O 1,26
%A _Labos Elemer_, Aug 09 2000
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