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 1-3 of 3 results.

A127989 a(n) = 2*n^3 - 2*n + 9.

Original entry on oeis.org

9, 21, 57, 129, 249, 429, 681, 1017, 1449, 1989, 2649, 3441, 4377, 5469, 6729, 8169, 9801, 11637, 13689, 15969, 18489, 21261, 24297, 27609, 31209, 35109, 39321, 43857, 48729, 53949, 59529, 65481, 71817, 78549, 85689, 93249, 101241, 109677, 118569
Offset: 1

Views

Author

Artur Jasinski, Feb 10 2007

Keywords

Comments

All numbers generated by the irreducible polynomial 2*n^3-2*n+9 are composite (indeed, they are multiples of 3). Largest prime factor A127990, smallest prime factor A127991, number of prime factors A127992.

Links

Crossrefs

Cf. A127990, A127991, A127992. Subsequence of A017629.

Programs

  • Magma
    [2*(n-1)*n*(n+1)+9: n in [1..50]]; // Vincenzo Librandi, Sep 13 2011
  • Mathematica
    Table[2x^3 - 2x + 9, {x, 1, 100}]
LinearRecurrence[{4,-6,4,-1}, {9, 21, 57, 129}, 50] (* G. C. Greubel, May 08 2018 *)
  • PARI
    a(n)=2*n^3-2*n+9 \\ Charles R Greathouse IV, Sep 30 2015

Formula

a(n) = 2*(n-1)*n*(n+1)+9 = 3*(2*A007290(n+1)+3).
G.f.: 3*x*(3-5*x+9*x^2-3*x^3)/(1-x)^4. - Bruno Berselli, Sep 09 2011
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Oct 11 2021

A127992 Number of distinct prime factors of 2*n^3 - 2*n + 9.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Feb 10 2007

Keywords

Comments

All numbers generated by the irreducible polynomial 2n^3 - 2n + 9 (A127989) are odd multiples of 3.

Links

Crossrefs

Cf. A001221, A127989, A127990 (largest prime factor), A127992 (number of prime factors).

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, 2n^3 - 2n + 9], {n, 1, 300}]; b = {}; Do[c = FactorInteger[a[[n]]]; d = Length[c]; AppendTo[b, d], {n, 1, Length[a]}]; b
Table[PrimeNu[2n^3-2n+9],{n,120}] (* Harvey P. Dale, Sep 24 2021 *)
  • PARI
    a(n)=omega(2*n^3 - 2*n + 9) \\ Charles R Greathouse IV, Mar 13 2020

Formula

a(n) = A001221(A127989(n)). - Amiram Eldar, Mar 13 2020

A127991 2*n^3 - 2*n + 9 divided by 3*largest prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 11, 1, 3, 21, 39, 1, 31, 1, 1, 1, 7, 297, 9, 351, 1, 1, 19, 91, 1, 101, 141, 51, 33, 37, 49, 1, 299, 37, 1, 3, 39, 21, 1, 11, 1, 19, 1, 31, 1071, 351, 9, 43, 1, 481, 11, 511, 83, 3, 3, 69, 1, 1, 91, 1, 19, 187, 39, 219, 417, 553, 37, 1, 1, 1, 1369, 117, 693, 423, 31
Offset: 1

Views

Author

Artur Jasinski, Feb 10 2007

Keywords

Links

Crossrefs

Cf. A127979, A127990, A127992.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, 2n^3 - 2n + 9], {n, 1, 300}]; b = {}; Do[c = FactorInteger[a[[n]]]; d = c[[Length[c]]]; AppendTo[b, a[[n]]/(3 d[[1]])], {n, 1, Length[a]}]; b
  • PARI
    gpf(n)=my(f=factor(n)[,1]); if(n==1,1,f[#f]);
a(n)=my(m=2*n^3-2*n+9); m/gpf(m)/3 \\ Charles R Greathouse IV, Mar 13 2020

Formula

a(n) = A127989(n)/(3*A006530(A127989(n))). - Amiram Eldar, Mar 13 2020
Showing 1-3 of 3 results.

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