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%I #11 Apr 23 2024 12:58:34
%S -81,3,89,177,267,359,453,549,647,747,849,953,1059,1167,1277,1389,
%T 1503,1619,1737,1857,1979,2103,2229,2357,2487,2619,2753,2889,3027,
%U 3167,3309,3453,3599,3747,3897,4049,4203,4359,4517,4677,4839,5003,5169,5337,5507,5679,5853,6029,6207,6387,6569
%N a(n) = n^2 + 83*n - 81.
%C Euler observed that the polynomial n^2 + n + 41 takes distinct prime values for the 40 consecutive integers from n = 0 to n = 39.
%C 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.
%C 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).
%C |a(3*n+2)| takes distinct prime values for the 24 consecutive integers from n = -14 to n = 9.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f. (163*x^2 - 246*x + 81)/(x - 1)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = -81, a(1) = 3 and a(2) = 89.
%F 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
%e For integer n in the interval [-41, 31], the unsigned sequence terms |a(n)| factorize as:
%e [ 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].
%p seq(n^2 + 83*n - 81, n = 0..50)
%t Table[n^2 + 83*n - 81, {n, 0, 50}]
%o (PARI) vector(50, n, n^2 + 83*n - 81)
%Y Cf. A005846, A007641, A048988, A366457.
%K sign,easy
%O 0,1
%A _Peter Bala_, Oct 12 2023