OFFSET
0,1
COMMENTS
Euler observed that the polynomial n^2 + n + 41 takes distinct prime values for the 40 consecutive integers from n = 0 to n = 39.
For the 73 integers in the interval -41 <= n <= 31, the unsigned sequence term |a(n)| is either a prime, (3^k)*prime (for some small value of k), or a power of 3 (for two values of n). See the example section below.
For the 88 integers in the interval -58 <= n <= 29, the unsigned sequence term |(1/3)*a(3*n+1)| = |3*n^2 + 85*n + 1| is either a prime, (3^k)*prime (for some small value of k), or a power of 3 (for two values of n).
|a(3*n+2)| takes distinct prime values for the 24 consecutive integers from n = -14 to n = 9.
LINKS
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f. (163*x^2 - 246*x + 81)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = -81, a(1) = 3 and a(2) = 89.
Sum_(n>=0) 1/a(n) = (psi((83+sqrt 7213)/2-psi((83-sqrt 7213)/2)/sqrt(7213) = 0.37949155... - R. J. Mathar, Apr 23 2024
EXAMPLE
For integer n in the interval [-41, 31], the unsigned sequence terms |a(n)| factorize as:
[ 3*601, 1801, 3*599, (3^2)*199, 1783, (3^2)*197, 3*587, 1747, 3*577, 3*571, 1693, 3*557, (3^3)*61, 1621, (3^3)*59, 3*521, 1531, 3*499, 3*487, 1423, 3*461, (3^2)*149, 1297, (3^2)*139, 3*401, 1153, 3*367, 3*349, 991, 3*311, (3^2)*97, 811, (3^2)*83, 3*227, 613, 3*181, 3*157, 397, 3*107, (3^5), 163, (3^4), 3, 89, 3*59, 3*89, 359, 3*151, (3^2)*61, 647, (3^2)*83, 3*283, 953, 3*353, 3*389, 1277, 3*463, (3^2)*167, 1619, (3^2)*193, 3*619, 1979, 3*701, 3*743, 2357, 3*829, (3^3)*97, 2753, (3^3)*107, 3*1009, 3167, 3*1103, 3*1151].
MAPLE
seq(n^2 + 83*n - 81, n = 0..50)
MATHEMATICA
Table[n^2 + 83*n - 81, {n, 0, 50}]
PROG
(PARI) vector(50, n, n^2 + 83*n - 81)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Oct 12 2023
STATUS
approved