A376554 -id:A376554 - cp's OEIS frontend

A376553 Largest unitary square divisor of binomial(n, floor(n/2)).

1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 1, 9, 9, 1, 4, 1, 4, 4, 1, 1, 4, 100, 25, 100, 25, 9, 144, 9, 9, 1, 4, 25, 100, 100, 25, 9, 36, 4, 1, 4, 1, 25, 400, 225, 900, 1764, 441, 196, 49, 49, 784, 4, 1, 1, 16, 1, 16, 16, 1, 441, 441, 49, 196, 49, 196, 36, 9

Offset: 0

Amiram Eldar, Sep 28 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A001405, A350388, A376554, A376555, A376556.

f[p_, e_] := If[EvenQ[e], p^e, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[Binomial[n, Floor[n/2]]]; Array[a, 100, 0]

a(n) = {my(f = factor(binomial(n, n\2))); prod(i = 1, #f~, if(f[i, 2]%2, 1, f[i, 1]^f[i, 2]));}

a(n) = A350388(A001405(n)).

a(n) = A376554(n)^2.

A376555 The number of unitary square divisors of binomial(n, floor(n/2)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 28 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001405, A056061, A056624, A376553, A376554, A376556.

f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[Binomial[n, Floor[n/2]]]; Array[a, 100, 0]

a(n) = vecprod(apply(x -> 1 << (1 - x%2), factor(binomial(n, n\2))[, 2]));

a(n) = A056624(A001405(n)).

A376556 The number of non-unitary square divisors of binomial(n, floor(n/2)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 28 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001405, A056061, A056626, A376553, A376554, A376555.

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); a[0] = a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[Binomial[n, Floor[n/2]]]) - Times @@ f2 @@@ fct; Array[a, 60, 0]

a(n) = {my(e = factor(binomial(n, n\2))[, 2]); vecprod(apply(x -> x\2 + 1, e)) - vecprod(apply(x -> 1 << (1 - x%2), e));}

a(n) = A056626(A001405(n)).

A056646 a(n) = A056622(A001405(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 2, 2, 1, 3, 3, 1, 2, 1, 2, 2, 1, 1, 2, 10, 5, 10, 5, 3, 12, 3, 3, 1, 2, 5, 10, 10, 5, 3, 6, 2, 1, 2, 1, 5, 20, 15, 30, 42, 21, 14, 7, 7, 28, 2, 1, 1, 4, 1, 4, 4, 2, 21, 21, 7, 14, 7, 14, 6, 3, 1, 2, 2, 1, 10, 5, 35, 140, 7, 14, 126, 63, 2, 1, 5, 20, 90, 45, 3, 12

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name "Square root of largest unitary square divisor of central binomial coefficient" was incorrect. See A376554 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056622(binomial(28,14)) = A056622(40116600) = 5.

Cf. A000188, A001405, A055229, A056056, A056059, A056622, A376554.

a(n) = A000188(A001405(n))/A055229(A001405(n)) = A056056(n)/A056059(n).

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

cp's OEIS Frontend

A376553 Largest unitary square divisor of binomial(n, floor(n/2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A376555 The number of unitary square divisors of binomial(n, floor(n/2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A376556 The number of non-unitary square divisors of binomial(n, floor(n/2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A056646 a(n) = A056622(A001405(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions