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%I #20 Apr 24 2024 05:36:14
%S 58,59,62,67,74,83,94,107,122,139,158,179,202,227,254,283,314,347,382,
%T 419,458,499,542,587,634,683,734,787,842,899,958,1019,1082,1147,1214,
%U 1283,1354,1427,1502,1579,1658,1739,1822,1907,1994,2083,2174,2267,2362,2459,2558,2659,2762,2867,2974,3083
%N a(n) = n^2 + 58.
%C Leonhard Euler observed that the polynomial n^2 + n + 41 takes distinct prime values for the 40 consecutive integers n = 0 to 39. Legendre showed that the first 29 terms of 2*n^2 + 29 (n = 0 to 28) are primes.
%C For even n = 2*m we have a(n) = 2*(2*m^2 + 29). It follows that a(n) is double a prime for the 29 even values of n in the integer interval [0, 57]. Calculation shows that a(n) takes distinct prime values for the 29 odd values of n in the interval [0, 57], except for a(29) = 29*31, a(33) = 31*37, a(41) = 37*47 and a(53) = 47*61. See the example section below.
%C The polynomial n^2 + 232 has similar properties. See A048988.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerPrime.html">Euler Prime</a>
%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.: (59*x^2 - 115*x + 58)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 58, a(1) = 59 and a(2) = 62.
%F Sum_{n>=0} 1/a(n) = (1+sqrt(58)*Pi*coth(sqrt(58)*Pi))/116 = 0.2148763... - _R. J. Mathar_, Apr 24 2024
%e The sequence terms factorized for 0 <= n <= 57:
%e [2*29, 59, 2*31, 67, 2*37, 83, 2*47, 107, 2*61, 139, 2*79, 179, 2*101, 227, 2*127, 283, 2*157, 347, 2*191, 419, 2*229, 499, 2*271, 587, 2*317, 683, 2*367, 787, 2*421, (29*31), 2*479, 1019, 2*541, (31*37), 2*607, 1283, 2*677, 1427, 2*751, 1579, 2*829, (37*47), 2*911, 1907, 2*997, 2083, 2*1087, 2267, 2*1181, 2459, 2*1279, 2659, 2*1381, (47*61), 2*1487, 3083, 2*1597, 3307].
%p seq(n^2 + 58, n = 0..50);
%t Table[n^2 + 58, {n, 0, 50}]
%o (PARI) vector(50, n, n^2 + 58)
%Y Cf. A005846, A007641, A048988, A330363, A366458.
%K nonn,easy
%O 0,1
%A _Peter Bala_, Oct 10 2023
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