cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A023828 Sum of exponents in prime-power factorization of C(4n,n-2).

Original entry on oeis.org

0, 3, 5, 5, 5, 9, 9, 11, 8, 8, 10, 12, 13, 15, 15, 16, 13, 16, 17, 16, 16, 18, 20, 22, 18, 20, 21, 20, 20, 22, 24, 25, 24, 25, 26, 26, 24, 28, 28, 29, 24, 28, 29, 29, 28, 29, 32, 34, 31, 31, 33, 31, 31, 34, 32, 34, 30, 33, 34, 37, 37, 40, 43, 42, 37, 39, 40, 41, 41, 41, 43, 44, 40, 45, 46, 46, 45, 47
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A004332, A023826, A023827, A023829, A023830, A023831, A023832, A023833, A023834, A023835, A023836.

Programs

  • Mathematica
    Join[{0}, Table[Total[FactorInteger[Binomial[4 n, n - 2]][[All, 2]]], {n, 3, 78}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[4*n, n-2]]; Array[a, 100, 2] (* Amiram Eldar, Jun 13 2025 *)
  • PARI
    a(n) = bigomega(binomial(4*n,n-2)); \\ Amiram Eldar, Jun 13 2025

Formula

From Amiram Eldar, Jun 13 2025 (Start)
a(n) = A001222(A004332(n)).
a(n) = A023827(n) - A001222(3*n+2) + A001222(n-1). (End)

Extensions

Offset corrected to 2 by Ivan Neretin, Nov 02 2017

A023829 Sum of exponents in prime-power factorization of C(4n,n-3).

Original entry on oeis.org

0, 4, 3, 5, 6, 8, 9, 9, 6, 10, 10, 13, 11, 15, 14, 15, 13, 17, 14, 17, 15, 19, 20, 18, 18, 21, 19, 21, 17, 24, 23, 26, 22, 26, 25, 25, 24, 28, 27, 26, 25, 28, 27, 29, 26, 31, 31, 33, 29, 34, 28, 31, 30, 33, 33, 32, 30, 34, 35, 37, 34, 42, 41, 41, 37, 40, 38, 42, 37, 44, 42, 41, 42, 45, 45, 46, 43, 45
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A004333, A023826, A023827, A023828, A023830, A023831, A023832, A023833, A023834, A023835, A023836.

Programs

  • Mathematica
    Table[PrimeOmega[Binomial[4 n, n - 3]], {n, 3, 78}] (* Ivan Neretin, Nov 02 2017 *)
  • PARI
    a(n) = bigomega(binomial(4*n,n-3)); \\ Amiram Eldar, Jun 13 2025

Formula

From Amiram Eldar, Jun 13 2025 (Start)
a(n) = A001222(A004333(n)).
a(n) = A023828(n) - A001222(3*n+3) + A001222(n-2). (End)

Extensions

Offset corrected to 3 by Ivan Neretin, Nov 02 2017

A023830 Sum of exponents in prime-power factorization of C(4n,n-4).

Original entry on oeis.org

0, 3, 4, 6, 6, 10, 8, 8, 8, 11, 12, 12, 13, 14, 15, 16, 12, 16, 15, 17, 18, 21, 17, 20, 19, 19, 22, 19, 21, 25, 25, 26, 23, 25, 25, 26, 26, 28, 25, 27, 25, 29, 28, 27, 31, 32, 31, 33, 30, 30, 31, 31, 31, 34, 32, 33, 32, 35, 35, 37, 39, 42, 42, 41, 37, 40, 41, 38, 42, 44, 40, 46, 42, 45, 46, 45, 44
Offset: 4

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A004334, A023826, A023827, A023828, A023829, A023831, A023832, A023833, A023834, A023835, A023836.

Programs

  • Maple
    with(numtheory): A023830:=n->bigomega(binomial(4*n,n-4)): seq(A023830(n), n=4..100); # Wesley Ivan Hurt, Oct 13 2014
  • Mathematica
    Join[{0},Table[Total[Transpose[FactorInteger[Binomial[4n,n-4]]][[2]]],{n,5,80}]] (* Harvey P. Dale, Oct 12 2014 *)
a[n_] := PrimeOmega[Binomial[4*n, n-4]]; Array[a, 100, 4] (* Amiram Eldar, Jun 13 2025 *)
  • PARI
    a(n) = bigomega(binomial(4*n, n-4)); \\ Michel Marcus, Oct 12 2014

Formula

a(n) = Omega( C(4n,n-4) ) = A001222( C(4n,n-4) ), n > 3.
From Amiram Eldar, Jun 13 2025 (Start)
a(n) = A001222(A004334(n)).
a(n) = A023829(n) - A001222(3*n+4) + A001222(n-3). (End)

Extensions

Offset changed by Wesley Ivan Hurt, Oct 13 2014

A023831 Sum of exponents in prime-power factorization of C(4n,n+1).

Original entry on oeis.org

2, 4, 4, 7, 7, 8, 8, 11, 11, 12, 8, 12, 13, 15, 13, 18, 14, 18, 14, 18, 20, 18, 17, 21, 22, 20, 19, 22, 20, 24, 19, 26, 24, 26, 23, 28, 27, 27, 27, 30, 29, 31, 27, 30, 30, 30, 27, 34, 31, 33, 31, 34, 33, 33, 30, 34, 33, 36, 31, 37, 38, 39, 37, 42, 39, 41, 38, 42, 43, 43, 38, 46, 44, 44, 44, 46, 46
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A023826, A023827, A023828, A023829, A023830, A023832, A023833, A023834, A023835, A023836.

Programs

  • Mathematica
    Table[PrimeOmega[Binomial[4 n, n + 1]], {n, 77}] (* Ivan Neretin, Nov 08 2017 *)
  • PARI
    a(n) = bigomega(binomial(4*n,n+1)); \\ Amiram Eldar, Jun 13 2025

Formula

a(n) = A023826(n) - A001222(3*n) + A001222(n+1). - Amiram Eldar, Jun 13 2025

A023832 Sum of exponents in prime-power factorization of C(4n,n+2).

Original entry on oeis.org

2, 3, 6, 6, 8, 6, 9, 10, 12, 10, 12, 12, 13, 12, 15, 16, 16, 16, 16, 17, 21, 16, 18, 20, 21, 19, 23, 20, 21, 20, 20, 26, 25, 23, 26, 27, 28, 24, 29, 29, 30, 31, 31, 29, 31, 26, 29, 33, 31, 31, 34, 32, 33, 31, 31, 33, 35, 33, 35, 36, 38, 35, 38, 40, 40, 39, 41, 41, 44, 40, 40, 46, 43, 43, 48, 44, 48
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A023826, A023827, A023828, A023829, A023830, A023831, A023833, A023834, A023835, A023836.

Programs

  • Mathematica
    Table[Total[Transpose[FactorInteger[Binomial[4n,n+2]]][[2]]],{n,80}] (* Harvey P. Dale, Jun 01 2014 *)
a[n_] := PrimeOmega[Binomial[4*n, n+2]]; Array[a, 100] (* Amiram Eldar, Jun 13 2025 *)
  • PARI
    a(n) = bigomega(binomial(4*n,n+2)); \\ Amiram Eldar, Jun 13 2025

Formula

a(n) = A023831(n) - A001222(3*n-1) + A001222(n+2). - Amiram Eldar, Jun 13 2025

A023835 Sum of exponents in prime-power factorization of C(4n,2n-1).

Original entry on oeis.org

2, 4, 6, 7, 7, 9, 11, 12, 13, 13, 13, 15, 14, 16, 17, 18, 16, 18, 18, 20, 22, 22, 23, 25, 24, 24, 25, 24, 24, 28, 27, 29, 27, 30, 30, 32, 31, 31, 34, 32, 31, 33, 34, 37, 36, 35, 36, 40, 37, 37, 38, 39, 39, 42, 41, 42, 42, 42, 42, 43, 44, 46, 47, 48, 45, 46, 44, 45, 48, 49, 47, 50, 47, 49, 53, 52, 51
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A023826, A023827, A023828, A023829, A023830, A023831, A023832, A023834, A023836, A076191.

Programs

  • Mathematica
    Table[PrimeOmega[Binomial[4 n, 2 n - 1]], {n, 78}] (* Ivan Neretin, Nov 08 2017 *)
  • PARI
    a(n) = bigomega(binomial(4*n,2*n-1)); \\ Amiram Eldar, Jun 13 2025

Formula

a(n) = A023832(n) - A001222(2*n+1) + A001222(2*n) = A023832(n) - A076191(2*n). - Amiram Eldar, Jun 13 2025

Extensions

a(74)-a(77) corrected by Ivan Neretin, Nov 08 2017

A023833 Sum of exponents in prime-power factorization of C(4n,n+3).

Original entry on oeis.org

0, 4, 5, 7, 6, 8, 8, 11, 11, 12, 11, 12, 10, 15, 13, 17, 15, 17, 16, 18, 18, 20, 17, 20, 19, 21, 21, 21, 18, 22, 20, 26, 22, 26, 25, 27, 25, 28, 28, 30, 29, 31, 30, 31, 28, 28, 27, 33, 30, 33, 31, 33, 30, 35, 30, 34, 33, 35, 36, 35, 33, 37, 37, 42, 38, 41, 39, 42, 41, 44, 39, 45, 42, 45, 46, 45, 44
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A023826, A023827, A023828, A023829, A023830, A023831, A023832, A023834, A023835, A023836.

Programs

  • Mathematica
    Join[{0},Total[Transpose[FactorInteger[#]][[2]]]&/@Table[Binomial[4n,n+3],{n,2,80}]] (* Harvey P. Dale, Dec 24 2014 *)
a[n_] := PrimeOmega[Binomial[4*n, n+3]]; Array[a, 100] (* Amiram Eldar, Jun 13 2025 *)
  • PARI
    a(n) = bigomega(binomial(4*n,n+3)); \\ Amiram Eldar, Jun 13 2025

Formula

a(n) = A023832(n) - A001222(3*n-2) + A001222(n+3). - Amiram Eldar, Jun 13 2025
Showing 1-7 of 7 results.

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