A056647 -id:A056647 - cp's OEIS frontend

A056648 a(n) = A034444(A056647(n)).

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 4, 2, 4, 2, 2, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 4, 2, 1, 2, 1, 2, 4, 4, 8, 8, 4, 4, 2, 2, 4, 2, 1, 1, 2, 1, 2, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 1, 2, 2, 1, 4, 2, 4, 8, 2, 4, 8, 4, 2, 1, 2, 4, 8, 4, 2, 4, 4, 8, 4, 4, 8, 16, 8, 4, 4, 2, 1, 2, 2, 1

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name, "Number of unitary square divisors of central binomial coefficient", was incorrect. See A376555 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A034444(A056647(28)) = A034444(25) = 2.

Cf. A000188, A001221, A001405, A008833, A034444, A055229, A056056, A056057, A056059, A056061, A376555.

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[2^(PrimeNu[ Sqrt[A008833[Binomial[n, Floor[n/2]]]]/A055229[Binomial[n, Floor[n/2]]]]), {n, 1, 25}] (* G. C. Greubel, May 20 2017 *)

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

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

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

Original entry on oeis.org

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

Labos Elemer, Aug 09 2000

nonn

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

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

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

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 *)

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

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

cp's OEIS Frontend

A056648 a(n) = A034444(A056647(n)).

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