A366457 a(n) = n^2 + 58.

58, 59, 62, 67, 74, 83, 94, 107, 122, 139, 158, 179, 202, 227, 254, 283, 314, 347, 382, 419, 458, 499, 542, 587, 634, 683, 734, 787, 842, 899, 958, 1019, 1082, 1147, 1214, 1283, 1354, 1427, 1502, 1579, 1658, 1739, 1822, 1907, 1994, 2083, 2174, 2267, 2362, 2459, 2558, 2659, 2762, 2867, 2974, 3083

Offset

0,1

Comments

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.

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.

The polynomial n^2 + 232 has similar properties. See A048988.

Links

Table of n, a(n) for n=0..55.

Eric Weisstein's World of Mathematics, Euler Prime

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: (59*x^2 - 115*x + 58)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 58, a(1) = 59 and a(2) = 62.

Sum_{n>=0} 1/a(n) = (1+sqrt(58)*Pi*coth(sqrt(58)*Pi))/116 = 0.2148763... - R. J. Mathar, Apr 24 2024

Example

The sequence terms factorized for 0 <= n <= 57:
[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].

Maple

seq(n^2 + 58, n = 0..50);

Mathematica

Table[n^2 + 58, {n, 0, 50}]

Prog

(PARI) vector(50, n, n^2 + 58)

Crossrefs

Cf. A005846, A007641, A048988, A330363, A366458.

Sequence in context: A225415 A033378 A259082 * A105326 A247096 A276186

Adjacent sequences: A366454 A366455 A366456 * A366458 A366459 A366460

Keyword

nonn,easy

Author

Peter Bala, Oct 10 2023

Status

approved