A212325 Prime-generating polynomial: a(n) = n^2 + 3*n - 167.

-167, -163, -157, -149, -139, -127, -113, -97, -79, -59, -37, -13, 13, 41, 71, 103, 137, 173, 211, 251, 293, 337, 383, 431, 481, 533, 587, 643, 701, 761, 823, 887, 953, 1021, 1091, 1163, 1237, 1313, 1391, 1471, 1553, 1637, 1723, 1811, 1901, 1993, 2087, 2183, 2281, 2381

Offset: 0

Author

Marius Coman, May 14 2012

Keywords

• sign
• easy

Comments

The polynomial generates 24 primes in absolute value (23 distinct ones) in row starting from n=0 (and 42 primes in absolute value for n from 0 to 46).

The polynomial n^2 - 49*n + 431 generates the same primes in reverse order.

Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 677) the polynomial 13*n^2 - 311*n + 1847 (13*n^2 - 469*n + 4217) generating 23 primes and two noncomposite numbers (in absolute value) in row starting from n=0 (1847, 1549, 1277, 1031, 811, 617, 449, 307, 191, 101, 37, -1, -13, 1, 41, 107, 199, 317, 461, 631, 827, 1049, 1297, 1571, 1871).

Note: another interesting algorithm to produce prime-generating polynomials could be N = m*n^2 + (6*m+1)*n + 8*m + 3, where m, 6*m+1 and 8*m+3 are primes. For m=7 then n=t-20 we get N = 7*t^2 - 237*t + 1999, which generates the following primes: 239, 163, 101, 53, 19, -1, -7, 1, 23, 59, 109, 173, 251 (we can see the same pattern: …, -1, -m, 1, …).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A060566 (an 80 primes generating pol.), A202018 (Euler's p.g.p.), A050268, A181963, A181973, A182409, A211773, A318791, A320772, A330363 (other p.g.p.).

Programs

Magma

[n^2+3*n-167: n in [0..47]]; // Bruno Berselli, May 18 2012

Mathematica

Table[n^2+3n-167,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{-167,-163,-157},50] (* Harvey P. Dale, Feb 08 2020 *)

PARI

Vec((-167+338*x-169*x^2)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

PARI

apply( {A212325(n)=(n+3)*n-167}, [0..55]) \\ M. F. Hasler, Feb 11 2025

Python

def A212325(n=None, upto=None): return(A212325(i)for i in range(n or 0, upto or 2**63)) if upto or n is None else(n+3)*n-167 # M. F. Hasler, Feb 11 2025

Formula

G.f.: (-167 + 338*x - 169x^2)/(1-x)^3. - Bruno Berselli, May 18 2012

From Elmo R. Oliveira, Feb 10 2025: (Start)

E.g.f.: exp(x)*(-167 + 4*x + x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Extensions

Edited by Bruno Berselli, May 18 2012