A259552 a(n) = (1/4)*n^4 - (1/2)*n^3 + (3/4)*n^2 - (1/2)*n + 41.

41, 43, 53, 83, 151, 281, 503, 853, 1373, 2111, 3121, 4463, 6203, 8413, 11171, 14561, 18673, 23603, 29453, 36331, 44351, 53633, 64303, 76493, 90341, 105991, 123593, 143303, 165283, 189701, 216731, 246553, 279353, 315323, 354661, 397571, 444263, 494953

Offset: 1

Author

Robert Potter, Jun 30 2015

Keywords

• nonn
• easy

Comments

Empirical Observation: Reasonably productive (better than 85% in first 24 terms) prime-generating polynomial.

All integers generated by this polynomial for 0 < n <= 24 are prime with the exception of a(14) = 47*179, a(17) = 71*263, and a(20) = 47*773.

Negative and zero values of n also produce primes but they are not unique.

a(n) = (1/4)*n^4 - (1/2)*n^3 + (3/4)*n^2 - (1/2)*n + 809, for 0 < n <= 24, is also reasonably productive but produces composites at a(4), a(7), a(19) and a(20).

a(n) = (1/4)*n^4 - (1/2)*n^3 + (3/4)*n^2 - (1/2)*n + 641, for 0 < n <= 24, is also quite productive.

Links

• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A202018.

Programs

Magma

[(1/4)*n^4-(1/2)*n^3+(3/4)*n^2-(1/2)*n+41: n in [1..40]]; // Vincenzo Librandi, Jul 03 2015

Maple

A259552:=n->n^4/4-n^3/2+3*n^2/4-n/2+41: seq(A259552(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2015

Mathematica

f[n_] := n^4/4 - n^3/2 + 3 n^2/4 - n/2 + 41; Array[f, 38] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {41, 43, 53, 83, 151}, 38] (* Robert G. Wilson v, Jul 07 2015 *)

Formula

G.f.: x*(41 - 162*x + 248*x^2 - 162*x^3 + 41*x^4)/(1-x)^5. - Vincenzo Librandi, Jul 03 2015

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n>5. - Wesley Ivan Hurt, Jul 09 2015

Extensions

Corrected and extended by Vincenzo Librandi, Jul 03 2015