A117624 Primes of the form f(k) = 9*k^6 - 804*k^5 + 29836*k^4 - 588615*k^3 + 6509950*k^2 - 38263500*k + 93363947 for values of k >= 0.

93363947, 61050823, 38620051, 23498297, 13649371, 7493947, 3835763, 1794301, 743947, 259631, 68947, 10753, 251, 547, 691, 197, 43, 151, 3347, 25801, 113947, 367883, 971251, 2227597, 4603211, 8776447, 15693523, 26630801, 102768947, 218611051, 1738931741

Offset

1,1

Comments

This polynomial f(n) generates 28 consecutive prime numbers for n = 0 to n = 27.

In n^2 + n + 41, substitute n -> 3*n^3 - 134*n^2 + 1980*n - 9663.

References

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2825

Carlos Rivera, Puzzle 232. Primes and Cubic polynomials, The Prime Puzzles & Problems Connection.

Eric Weisstein's World of Mathematics, Prime-generating polynomial.

Example

f(1) = 9(1)^6 - 804(1)^5 + 29836(1)^4 - 588615(1)^3 + 6509950(1)^2 - 38263500(1) + 93363947 = 61050823, a prime number.

Mathematica

f[n_] := 9n^6-804n^5+29836n^4-588615n^3+6509950n^2-38263500n+93363947; f[Select[Range[0, 100], PrimeQ[f[ # ]] &]] (* Stefan Steinerberger, Apr 16 2006 *)

Prog

(Magma) [a: n in [0..200] | IsPrime(a) where a is 9*n^6 - 804*n^5 + 29836*n^4 - 588615*n^3 + 6509950*n^2 - 38263500*n + 93363947 ]; // Vincenzo Librandi, Jul 28 2025

Crossrefs

Cf. A005846.

Sequence in context: A186051 A205665 A205465 * A147527 A293244 A136634

Adjacent sequences: A117621 A117622 A117623 * A117625 A117626 A117627

Keyword

easy,nonn

Author

Parviz Afereidoon (afereidoon(AT)gmail.com), Apr 08 2006

Extensions

Edited by Don Reble, Apr 14 2006

More terms from Petros Hadjicostas, Nov 04 2019

Status

approved