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 1-4 of 4 results.

A023847 Sum of exponents in prime-power factorization of binomial(5n, 2n).

Original entry on oeis.org

0, 2, 4, 4, 6, 8, 8, 11, 11, 11, 13, 13, 13, 18, 16, 17, 17, 19, 18, 18, 22, 24, 21, 23, 23, 24, 28, 26, 26, 30, 29, 32, 28, 30, 31, 31, 32, 35, 35, 36, 36, 36, 37, 33, 35, 38, 36, 39, 36, 40, 40, 41, 45, 48, 43, 46, 46, 45, 50, 47, 49, 52, 52, 49, 46, 51, 51, 50, 50, 55, 51, 57, 54, 57, 57, 55, 59, 62
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A001450, A022559.
Cf. A023816-A023854.

Programs

  • Mathematica
    Table[PrimeOmega[Binomial[5 n, 2 n]], {n, 77}] (* Ivan Neretin, Nov 09 2017 *)
  • PARI
    a(n) = bigomega(binomial(5*n, 2*n)); \\ Michel Marcus, Nov 09 2017
  • PARI
    a(n) = my(res = 0); forprime(p = 2, 5*n, res += (val(5*n, p) - val(2*n, p) - val(3*n, p))); res
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Nov 09 2017

Formula

a(n) = A001222(A001450(n)). - Michel Marcus, Nov 09 2017

Extensions

a(0) = 0 prepended by David A. Corneth, Nov 09 2017

A023852 Sum of exponents in prime-power factorization of binomial(6n, n).

Original entry on oeis.org

0, 2, 3, 6, 5, 7, 9, 9, 8, 10, 10, 14, 13, 12, 15, 16, 15, 16, 16, 19, 16, 18, 22, 24, 22, 23, 21, 23, 22, 22, 24, 25, 25, 24, 25, 28, 25, 27, 28, 28, 26, 31, 30, 38, 36, 33, 38, 38, 37, 35, 36, 38, 33, 33, 37, 38, 36, 39, 37, 43, 39, 41, 43, 46, 42, 40, 41, 45, 46, 43, 47, 46, 45, 46, 47, 49, 47, 45
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A004355, A022559.
Cf. A023853, A023854.

Programs

  • Maple
    with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(6*n,n)) fi end: seq(a(n), n=0..77); # Zerinvary Lajos, Apr 11 2008
  • Mathematica
    Table[PrimeOmega[Binomial[6 n, n]], {n, 0, 77}] (* Ivan Neretin, Nov 09 2017 *)
  • PARI
    a(n) = bigomega(binomial(6*n, n)); \\ Michel Marcus, Nov 10 2017

Formula

a(n) = A001222(A004355(n)). - Michel Marcus, Nov 10 2017
a(n) = A022559(6*n) - A022559(5*n) - A022559(n). - Amiram Eldar, Jun 11 2025

Extensions

a(0)=0 inserted by Amiram Eldar, Jun 11 2025

A023853 Sum of exponents in prime-power factorization of binomial(6n, 2n).

Original entry on oeis.org

0, 2, 4, 6, 6, 8, 11, 11, 10, 11, 13, 18, 17, 17, 17, 19, 18, 20, 19, 22, 22, 23, 27, 28, 27, 26, 29, 31, 29, 31, 32, 35, 31, 31, 31, 35, 33, 35, 37, 36, 36, 37, 39, 43, 42, 43, 46, 47, 43, 44, 45, 48, 47, 46, 50, 53, 51, 52, 53, 57, 55, 55, 56, 56, 51, 52, 54, 56, 56, 55, 58, 62, 58, 59, 60, 61, 62
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A001222, A022559, A259613.
Cf. A023852, A023854.

Programs

  • Mathematica
    Join[{0}, Table[Total[Transpose[FactorInteger[Binomial[6 n,2 n]]] [[2]]],{n,80}]] (* Harvey P. Dale, Jun 28 2011 *)
a[n_] := PrimeOmega[Binomial[6*n, 2*n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)
  • PARI
    a(n) = bigomega(binomial(6*n, 2*n)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(3*A259613(n)) for n >= 1.
a(n) = A022559(6*n) - A022559(4*n) - A022559(2*n). (End)

Extensions

a(0)=0 inserted by Amiram Eldar, Jun 11 2025

A023586 Exponent of least prime factor in prime factorization of 2*prime(n)-1.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

First differs from A023584 at n = 56. - Sean A. Irvine, Jun 07 2019

Links

Crossrefs

Cf. A023854.

Programs

  • PARI
    a(n) = factor(2*prime(n)-1)[1, 2]; \\ Michel Marcus, Jun 08 2019

Extensions

Title corrected by Sean A. Irvine, Jun 07 2019
Showing 1-4 of 4 results.

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