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A056649 Number of non-unitary square divisors of central binomial coefficient. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,26

LINKS

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

FORMULA

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))).

EXAMPLE

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.

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, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061.

Sequence in context: A080413 A004517 A254350 * A075242 A161489 A050975

Adjacent sequences:  A056646 A056647 A056648 * A056650 A056651 A056652

KEYWORD

nonn

AUTHOR

Labos Elemer, Aug 09 2000

STATUS

approved

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)