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.

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A214732 a(n) = 25*n^2 + 15*n + 1021.

Original entry on oeis.org

1021, 1061, 1151, 1291, 1481, 1721, 2011, 2351, 2741, 3181, 3671, 4211, 4801, 5441, 6131, 6871, 7661, 8501, 9391, 10331, 11321, 12361, 13451, 14591, 15781, 17021, 18311, 19651, 21041, 22481, 23971, 25511, 27101, 28741, 30431, 32171, 33961, 35801, 37691
Offset: 0

Views

Author

Robert Potter, Jul 27 2012

Keywords

Comments

This is the case m=5 and k=41 of the formula m^2*n^2 + (m^2 - 2*m)*n + (m^2*k) - (m-1). The most famous example is when m=1 and k=41 (Euler's generating polynomial). With k=41 the formula gives consecutive primes for m=10 and n=0..10, m=17 and n=0..10, m=86 and n=0..8. It is interesting to note that the sequences produced are all factors of the semiprimes produced by m=1, k=41. The other famous values to try for k are 5, 11 and 17 as these all produce primes up to k^2.

Links

Crossrefs

Cf. A215814.

Programs

Formula

G.f.: (1021-2002*x+1031*x^2)/(1-x)^3. - Bruno Berselli, Aug 28 2012
E.g.f.: (1021 + 40*x + 25*x^2)*exp(x). - G. C. Greubel, Apr 26 2021

A258841 a(n) = 9*n^2 - 237*n + 1927.

Original entry on oeis.org

1927, 1699, 1489, 1297, 1123, 967, 829, 709, 607, 523, 457, 409, 379, 367, 373, 397, 439, 499, 577, 673, 787, 919, 1069, 1237, 1423, 1627, 1849, 2089, 2347, 2623, 2917, 3229, 3559, 3907, 4273, 4657, 5059, 5479, 5917, 6373, 6847, 7339, 7849, 8377, 8923, 9487, 10069
Offset: 0

Views

Author

Robert Potter, Jun 12 2015

Keywords

Comments

Empirical observation. All integers generated by polynomial for 0 < n <= 37 are prime with the exception of a(26) = 43^2 and a(29) = 43*61.

Links

Crossrefs

Cf. A202018, A214732, A215814.

Programs

  • Magma
    [9*n^2-237*n+1927: n in [0..50]]; // Vincenzo Librandi, Jun 22 2015
  • Mathematica
    Table[9 n^2 - 237 n + 1927, {n, 0, 25}] (* Michael De Vlieger, Jun 12 2015 *)
LinearRecurrence[{3,-3,1},{1927,1699,1489},50] (* Harvey P. Dale, Oct 08 2024 *)
  • PARI
    vector(50, n, 9*n^2 - 237*n + 1927) \\ Michel Marcus, Jun 21 2015

Formula

From Vincenzo Librandi, Jun 22 2015: (Start)
G.f.: (1927 - 4082*x + 2173*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(1927 - 228*x + 9*x^2). - Elmo R. Oliveira, Feb 09 2025
Showing 1-2 of 2 results.

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