A121887 a(n) = (n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316)/4.

-1705829, -1313701, -991127, -729173, -519643, -355049, -228581, -134077, -65993, -19373, 10181, 26539, 33073, 32687, 27847, 20611, 12659, 5323, -383, -3733, -4259, -1721, 3923, 12547, 23887, 37571, 53149, 70123, 87977, 106207, 124351, 142019, 158923, 174907, 189977, 204331, 218389

Offset: 0

Author

Roger L. Bagula, Aug 31 2006

Keywords

• sign
• easy

Comments

Prime generating polynomial found by Shyam Sunder Gupta. The first 57 values (n=0..56) are primes.

In fact, this polynomial was first found by F. Dress and B. Landreau in 2002 and not by Gupta. See, e.g., Ribenboim's book, page 148. - Hugo Pfoertner, Dec 12 2019

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Ed Pegg Jr., Prime generating polynomial, July 17, 2006.
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A330363 for a polynomial improving the record to 58 consecutive primes.

Programs

GAP

List([0..35], n-> (n^5 -133*n^4 +6729*n^3 -158379*n^2 +1720294*n -6823316)/4); # G. C. Greubel, Oct 07 2019

Magma

[(n^5 -133*n^4 +6729*n^3 -158379*n^2 +1720294*n -6823316)/4: n in [0..35]]; // G. C. Greubel, Oct 07 2019

Maple

seq((n^5 -133*n^4 +6729*n^3 -158379*n^2 +1720294*n -6823316)/4, n=0..35); # G. C. Greubel, Oct 07 2019

Mathematica

Table[(n^5 -133*n^4 +6729*n^3 -158379*n^2 +1720294*n -6823316)/4, {n, 0, 35}]

PARI

vector(35, n, my(m=n-1); (m^5 -133*m^4 +6729*m^3 -158379*m^2 +1720294*m -6823316)/4) \\ G. C. Greubel, Oct 07 2019

Sage

[(n^5 -133*n^4 +6729*n^3 -158379*n^2 +1720294*n -6823316)/4 for n in (0..35)] # G. C. Greubel, Oct 07 2019

Formula

G.f.: (-1705829 + 8921273*x - 18696356*x^2 + 19628654*x^3 - 10324925*x^4 + 2177213*x^5)/(1-x)^6. - R. J. Mathar, Sep 13 2011

E.g.f.: (-6823316 + 1568512 x - 139108 x^2 + 5956 x^3 - 123 x^4 + x^5)*exp(x)/4. - G. C. Greubel, Oct 07 2019

Extensions

Edited by N. J. A. Sloane, Sep 05 2006