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 1-8 of 8 results.

A023988 Duplicate of A023819.

Original entry on oeis.org

1, 2, 4, 4, 4, 6, 8, 6, 7, 8, 11, 11, 11, 11, 12, 10, 12, 11, 15, 13, 13, 18, 18, 17, 16, 17, 19, 17
Offset: 1

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Author

Keywords

A023987 Sum of exponents of primes in C(5n,3n) - sum of exponents of primes in C(3n,2n).

Original entry on oeis.org

0, 1, 2, 0, 2, 4, 2, 3, 5, 4, 5, 2, 2, 7, 5, 5, 7, 7, 7, 3, 9, 11, 3, 5, 6, 8, 11, 7, 9, 12, 10, 10, 10, 12, 11, 10, 13, 14, 13, 13, 14, 15, 14, 5, 8, 11, 8, 9, 9, 12, 14, 13, 16, 20, 12, 14, 17, 14, 19, 12, 17, 19, 17, 15, 15, 21, 20, 16, 19, 23, 16, 21, 21, 23, 22, 17, 22, 26, 24, 24, 26, 26, 24, 22, 23
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A022559, A023819, A023847.
Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Binomial[5*n, 3*n]] - PrimeOmega[Binomial[3*n, 2*n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega(binomial(5*n, 3*n)) - bigomega(binomial(3*n, 2*n)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023847(n) - A023819(n).
a(n) = A022559(5*n) - 2*A022559(3*n) + A022559(n). (End)

Extensions

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023821 Sum of exponents in prime-power factorization of C(3n,n-2).

Original entry on oeis.org

0, 2, 3, 3, 6, 5, 6, 8, 8, 9, 11, 9, 10, 10, 12, 11, 12, 13, 13, 14, 17, 15, 17, 17, 16, 18, 19, 17, 20, 17, 19, 21, 19, 19, 22, 20, 20, 21, 22, 21, 23, 26, 26, 28, 29, 25, 29, 28, 27, 28, 29, 26, 32, 31, 31, 32, 30, 31, 33, 33, 31, 31, 35, 33, 32, 30, 31, 32, 36, 32, 35, 34, 33, 38, 37, 35, 38, 35
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A003408, A023819, A023820, A023822, A023823, A023824, A023825..

Programs

  • Mathematica
    Join[{0},Table[Total[Transpose[FactorInteger[Binomial[3n,n-2]]][[2]]],{n,3,80}]] (* Harvey P. Dale, Sep 23 2014 *)
a[n_] := PrimeOmega[Binomial[3*n, n-2]]; Array[a, 100, 2] (* Amiram Eldar, Jun 12 2025 *)
  • PARI
    a(n) = bigomega(binomial(3*n, n-2)); \\ Amiram Eldar, Jun 12 2025

Formula

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A003408(n-2)).
a(n) = A023820(n) - A001222(2*n+2) + A001222(n-1). (End)

A023822 Sum of exponents in prime-power factorization of C(3n,n-3).

Original entry on oeis.org

0, 3, 3, 6, 5, 7, 7, 10, 9, 10, 9, 12, 9, 12, 12, 14, 13, 15, 12, 19, 15, 17, 17, 18, 18, 20, 19, 20, 16, 21, 20, 23, 20, 21, 20, 23, 18, 23, 21, 25, 26, 27, 27, 30, 27, 28, 28, 31, 27, 31, 27, 33, 31, 33, 31, 32, 31, 33, 31, 34, 30, 36, 34, 34, 31, 33, 31, 37, 32, 35, 34, 37, 36, 37, 37, 39, 35, 37
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A004321, A023819, A023820, A023822, A023823, A023824, A023825.

Programs

  • Mathematica
    Join[{0}, Table[Total[FactorInteger[Binomial[3 n, n - 3]][[All, 2]]], {n, 4, 80}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[3*n, n-3]]; Array[a, 100, 3] (* Amiram Eldar, Jun 12 2025 *)
  • PARI
    a(n) = bigomega(binomial(3*n, n-3)); \\ Amiram Eldar, Jun 12 2025

Formula

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A004321(n)).
a(n) = A023821(n) - A001222(2*n+3) + A001222(n-2). (End)

Extensions

Offset corrected to 3 by Ivan Neretin, Nov 02 2017

A023823 Sum of exponents in prime-power factorization of C(3n,n+1).

Original entry on oeis.org

1, 3, 4, 6, 4, 8, 7, 8, 8, 10, 10, 14, 11, 12, 11, 14, 11, 14, 14, 15, 14, 20, 16, 20, 17, 17, 20, 20, 17, 22, 19, 22, 19, 21, 20, 23, 21, 23, 22, 26, 20, 26, 27, 28, 29, 30, 27, 31, 28, 28, 28, 32, 26, 34, 31, 32, 32, 33, 33, 36, 33, 35, 32, 36, 30, 34, 33, 33, 32, 38, 33, 38, 34, 35, 39, 39, 36, 39
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A023819, A023820, A023821, A023822, A023824, A023825.

Programs

  • Mathematica
    Table[Total[FactorInteger[Binomial[3 n, n + 1]][[All, 2]]], {n, 78}] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[3*n, n+1]]; Array[a, 100] (* Amiram Eldar, Jun 12 2025 *)
  • PARI
    a(n) = bigomega(binomial(3*n, n+1)); \\ Amiram Eldar, Jun 12 2025

Formula

a(n) = A023819(n) + A001222(2*n) - A001222(n+1). - Amiram Eldar, Jun 12 2025

Extensions

a(53) corrected by Ivan Neretin, Nov 02 2017

A023824 Sum of exponents in prime-power factorization of C(3n,n+2).

Original entry on oeis.org

0, 2, 4, 5, 5, 6, 6, 8, 8, 8, 11, 13, 11, 11, 11, 12, 12, 13, 13, 15, 14, 17, 17, 19, 16, 16, 20, 19, 18, 18, 18, 23, 19, 18, 21, 22, 20, 22, 23, 24, 23, 24, 26, 28, 29, 27, 27, 30, 27, 28, 28, 29, 27, 31, 30, 32, 32, 31, 35, 36, 32, 31, 33, 34, 31, 32, 33, 34, 32, 34, 34, 38, 33, 35, 38, 37, 38, 36, 33
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			For n = 4, C(12,6) = 924 = 2^2*3*7*11, so a(4) = 2+1+1+1 = 5.

Links

Crossrefs

Cf. A001222, A023819, A023820, A023821, A023822, A023824, A023825.

Programs

  • Mathematica
    Table[Total[FactorInteger[Binomial[3n,n+2]][[All,2]]],{n,2,80}] (* Harvey P. Dale, Dec 24 2016 *)
a[n_] := PrimeOmega[Binomial[3*n, n+2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2025 *)
  • PARI
    a(n) = bigomega(binomial(3*n, n+2)); \\ Amiram Eldar, Jun 12 2025

Formula

a(n) = A023823(n) + A001222(2*n-1) - A001222(n+2). - Amiram Eldar, Jun 12 2025

Extensions

a(1)=0 added by Ivan Neretin, Nov 02 2017

A023825 Sum of exponents in prime-power factorization of C(3n,n+3).

Original entry on oeis.org

2, 4, 6, 5, 6, 7, 9, 9, 10, 12, 13, 11, 12, 11, 14, 14, 13, 15, 16, 14, 18, 18, 18, 18, 18, 20, 22, 17, 18, 20, 23, 21, 20, 22, 23, 21, 23, 23, 26, 25, 23, 28, 29, 28, 29, 27, 30, 30, 30, 28, 30, 27, 31, 33, 34, 33, 33, 36, 35, 31, 31, 33, 37, 35, 33, 34, 35, 31, 36, 36, 37, 36, 35, 38, 40, 37, 35, 35
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A023819, A023821, A023822, A023823, A023824.

Programs

  • Mathematica
    Table[Total[FactorInteger[Binomial[3 n, n + 3]][[All, 2]]], {n, 2, 79}] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[3*n, n+3]]; Array[a, 100, 2] (* Amiram Eldar, Jun 12 2025 *)
  • PARI
    a(n) = bigomega(binomial(3*n, n+3)); \\ Amiram Eldar, Jun 12 2025

Formula

a(n) = A023824(n) + A001222(2*n-2) - A001222(n+3). - Amiram Eldar, Jun 12 2025

Extensions

a(2)=2 added and offset corrected to 2 by Ivan Neretin, Nov 02 2017

A023820 Sum of exponents in prime-power factorization of C(3n,n-1).

Original entry on oeis.org

0, 2, 4, 4, 4, 7, 7, 8, 8, 8, 11, 12, 9, 12, 13, 12, 11, 13, 14, 15, 14, 17, 18, 19, 16, 18, 20, 18, 18, 21, 20, 21, 19, 20, 22, 22, 19, 22, 24, 22, 21, 24, 27, 29, 28, 28, 29, 31, 27, 28, 29, 29, 28, 34, 32, 32, 31, 30, 34, 34, 32, 34, 36, 35, 31, 32, 31, 33, 33, 36, 35, 36, 32, 36, 40, 37, 36, 38
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A004319, A023819, A023821, A023822, A023823, A023824, A023825.

Programs

  • Mathematica
    Join[{0}, Table[Total[FactorInteger[Binomial[3 n, n - 1]][[All, 2]]], {n, 2, 78}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[3*n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 12 2025 *)
  • PARI
    a(n) = bigomega(binomial(3*n, n-1)); \\ Amiram Eldar, Jun 12 2025

Formula

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A004319(n)).
a(n) = A023819(n) - A001222(2*n+1) + A001222(n). (End)
Showing 1-8 of 8 results.

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