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 1-10 of 13 results. Next

A023978 Sum of exponents in prime-power factorization of multinomial coefficient M(3n; n,n,n).

Original entry on oeis.org

0, 2, 4, 7, 7, 9, 11, 14, 12, 13, 14, 19, 18, 20, 22, 23, 19, 23, 22, 27, 25, 25, 30, 33, 30, 30, 32, 33, 31, 34, 34, 38, 33, 34, 36, 38, 34, 37, 40, 42, 39, 42, 43, 50, 49, 48, 50, 54, 49, 50, 49, 51, 51, 53, 54, 55, 51, 53, 54, 61, 57, 60, 63, 63, 56, 56, 56, 61, 60, 61, 63, 66, 61, 64, 67, 69, 67, 68
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A006480, A022559.
Cf. A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((3*n)! / n!^3); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A006480(n)).
a(n) = A022559(3*n) - 3*A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023979 Sum of exponents in prime-power factorization of multinomial coefficient M(4n; n,n,n,n).

Original entry on oeis.org

0, 4, 7, 11, 12, 16, 17, 23, 21, 23, 24, 28, 27, 33, 36, 37, 33, 38, 37, 42, 41, 44, 46, 52, 48, 51, 52, 51, 50, 55, 55, 60, 55, 57, 61, 62, 58, 64, 66, 70, 65, 72, 71, 78, 78, 76, 78, 84, 79, 81, 80, 82, 82, 88, 84, 86, 82, 85, 88, 94, 90, 98, 102, 102, 93, 95, 94, 100, 100, 104, 103, 107, 102
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A008977, A022559.
Cf. A023978, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[n, n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((4*n)! / n!^4); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A008977(n)).
a(n) = A022559(4*n) - 4*A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023980 Sum of exponents in prime-power factorization of multinomial coefficient M(4n;2n,n,n).

Original entry on oeis.org

0, 3, 5, 8, 9, 11, 12, 17, 15, 17, 18, 20, 20, 24, 25, 26, 24, 27, 26, 30, 29, 32, 34, 37, 35, 37, 37, 37, 36, 39, 40, 44, 40, 41, 45, 45, 43, 48, 48, 51, 48, 51, 51, 56, 56, 55, 56, 60, 57, 59, 57, 59, 60, 63, 61, 63, 60, 63, 65, 68, 65, 71, 74, 73, 68, 69, 69, 73, 71, 75, 75, 77, 74, 78, 79, 81, 81
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A000897, A022559.
Cf. A023978, A023979, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[2*n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((4*n)! / ((2*n)! * n!^2)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A000897(n)).
a(n) = A022559(4*n) - A022559(2*n) - 2*A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023981 Sum of exponents in prime-power factorization of multinomial coefficient M(5n; n,n,n,n,n).

Original entry on oeis.org

0, 5, 10, 14, 16, 22, 24, 31, 29, 30, 33, 40, 38, 47, 49, 51, 45, 53, 51, 57, 59, 61, 63, 71, 66, 68, 75, 73, 71, 80, 78, 86, 76, 80, 83, 86, 81, 88, 93, 97, 92, 99, 100, 105, 106, 107, 108, 117, 107, 112, 112, 115, 118, 126, 120, 124, 119, 120, 127, 134, 131, 139, 143, 141, 127, 133
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A008978, A022559.
Cf. A023978, A023979, A023980, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[n, n, n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((5*n)! / n!^5); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A008978(n)).
a(n) = A022559(5*n) - 5*A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023982 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;3n,n,n).

Original entry on oeis.org

0, 3, 6, 7, 9, 13, 13, 17, 17, 17, 19, 21, 20, 27, 27, 28, 26, 30, 29, 30, 34, 36, 33, 38, 36, 38, 43, 40, 40, 46, 44, 48, 43, 46, 47, 48, 47, 51, 53, 55, 53, 57, 57, 55, 57, 59, 58, 63, 58, 62, 63, 64, 67, 73, 66, 69, 68, 67, 73, 73, 74, 79, 80, 78, 71, 77, 76, 77, 79, 84, 79, 87, 82, 87, 89, 86, 89
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A001451, A022559.
Cf. A023978, A023979, A023980, A023981, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

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

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001451(n)).
a(n) = A022559(5*n) - A022559(3*n) - 2*A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023983 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;2n,2n,n).

Original entry on oeis.org

0, 3, 6, 8, 10, 12, 14, 19, 17, 18, 21, 24, 24, 29, 27, 29, 27, 31, 29, 33, 35, 37, 39, 41, 40, 40, 45, 45, 43, 48, 48, 54, 46, 48, 51, 52, 51, 56, 57, 59, 58, 57, 60, 61, 62, 65, 64, 69, 63, 68, 66, 69, 74, 76, 74, 78, 75, 76, 81, 82, 81, 85, 87, 83, 77, 81, 82, 84, 81, 87, 86, 93, 87, 91, 92, 93, 96
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A001459, A022559.
Cf. A023978, A023979, A023980, A023981, A023982, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[2*n, 2*n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((5*n)! / ((2*n)!^2*n!)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001459(n)).
a(n) = A022559(5*n) - 2*A022559(2*n) - A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023984 Sum of exponents in prime-power factorization of multinomial coefficient M(6n; n,n,n,n,n,n).

Original entry on oeis.org

0, 7, 13, 20, 21, 29, 33, 40, 37, 40, 43, 54, 51, 59, 64, 67, 60, 69, 67, 76, 75, 79, 85, 95, 88, 91, 96, 96, 93, 102, 102, 111, 101, 104, 108, 114, 106, 115, 121, 125, 118, 130, 130, 143, 142, 140, 146, 155, 144, 147, 148, 153, 151, 159, 157, 162, 155, 159, 164, 177, 170, 180, 186
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A008979, A022559.
Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[n, n, n, n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((6*n)! / (n!)^6); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A008979(n)).
a(n) = A022559(6*n) - 6*A022559(n). (End)

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 5, 3, 5, 6, 4, 6, 6, 3, 4, 6, 5, 4, 6, 5, 8, 10, 7, 9, 9, 7, 9, 8, 7, 10, 12, 10, 9, 13, 11, 13, 16, 12, 13, 14, 9, 11, 12, 12, 13, 12, 12, 13, 15, 11, 13, 16, 13, 15, 17, 16, 19, 19, 16, 15, 17, 18, 15, 18, 17, 19, 19, 13, 17, 19, 17, 18, 18, 15, 19, 21, 18, 17, 20, 19, 19, 22, 19, 22
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A022559, A023816, A023834.
Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023987, A023990, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Binomial[4*n, 2*n]] - PrimeOmega[Binomial[2*n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = my(v = binomial(4*n, 2*n)/binomial(2*n, n)); bigomega(numerator(v)) - bigomega(denominator(v)); \\ Michel Marcus, Sep 30 2013
  • PARI
    vp(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(s);forprime(p=2,4*n,s+=vp(4*n,p)-3*vp(2*n,p)+2*vp(n,p)); s \\ Charles R Greathouse IV, Sep 30 2013

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023834(n) - A023816(n).
a(n) = A022559(4*n) - 3*A022559(2*n) + 2*A022559(n). (End)

Extensions

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

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

A023990 Sum of exponents of primes in multinomial coefficient M(4n; 2n,n,n) - sum of exponents of primes in M(3n; n,n,n).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 3, 3, 4, 4, 1, 2, 4, 3, 3, 5, 4, 4, 3, 4, 7, 4, 4, 5, 7, 5, 4, 5, 5, 6, 6, 7, 7, 9, 7, 9, 11, 8, 9, 9, 9, 8, 6, 7, 7, 6, 6, 8, 9, 8, 8, 9, 10, 7, 8, 9, 10, 11, 7, 8, 11, 11, 10, 12, 13, 13, 12, 11, 14, 12, 11, 13, 14, 12, 12, 14, 14, 13, 14, 14, 15, 15, 13, 14, 14, 11, 10, 13, 12, 13, 13, 12, 15
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A000897, A006480, A022559.
Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023991, A023992, A023993.

Programs

  • Mathematica
    a[n_] := PrimeOmega[Multinomial[2*n, n, n]] - PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega((4*n)!/((2*n)!*n!^2)) - bigomega((3*n)!/(n!^3)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023980(n) - A023978(n) = A001222(A000897(n)) - A001222(A006480(n)).
a(n) = A022559(4*n) + 2*A022559(n) - A022559(2*n) - A022559(3*n). (End)

Extensions

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025
Showing 1-10 of 13 results. Next

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