A259645 Numbers m such that m^2 + 1, 3*m - 1 and m^2 + m + 41 are all prime.

1, 2, 4, 6, 10, 14, 16, 20, 24, 36, 66, 90, 94, 116, 120, 134, 150, 156, 160, 206, 240, 280, 340, 350, 384, 396, 430, 436, 470, 536, 634, 690, 700, 714, 780, 826, 864, 930, 946, 960, 1004, 1124, 1150, 1176, 1294, 1316, 1376, 1410, 1430, 1494, 1644, 1674

Offset: 1

Author

Reinhard Zumkeller, Jul 03 2015

Keywords

• nonn

Comments

This sequence is infinite if the generalized Dickson's conjecture holds.

Examples

			.            | (i, j, k) such that |        corresponding
.            | a(n) = A005574(i)   |        prime triples
.     |      |      = A087370(j)   |        let m = a(n):
.   n | a(n) |      = A056561(k)   |  (m^2+1, 3*m-1, m^2+m+41)
.  ---+------+---------------------+--------------------------
.   1 |    1 |     (1,  1,  2)     |        (2,   2,   43)
.   2 |    2 |     (2,  2,  3)     |        (5,   5,   47)
.   3 |    4 |     (3,  3,  5)     |       (17,  11,   61)
.   4 |    6 |     (4,  4,  7)     |       (37,  17,   83)
.   5 |   10 |     (5,  6, 11)     |      (101,  29,  151)
.   6 |   14 |     (6,  7, 13)     |      (197,  41,  251)
.   7 |   16 |     (7,  8, 15)     |      (257,  47,  313)
.   8 |   20 |     (8, 10, 21)     |      (401,  59,  461)
.   9 |   24 |     (9, 11, 25)     |      (597,  71,  641)
.  10 |   36 |    (11, 15, 37)     |     (1297, 107, 1373)
.  11 |   66 |    (15, 24, 61)     |     (4357, 197, 4463)
.  12 |   90 |    (18, 31, 79)     |     (8101, 269, 8231)  .
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Wikipedia, Bunyakovsky conjecture
• Wikipedia, Dickson's conjecture

Crossrefs

Intersection of A005574, A087370 and A056561.

Programs

Haskell

import Data.List.Ordered (isect)
a259645 n = a259645_list !! (n-1)
a259645_list = a005574_list `isect` a087370_list `isect` a056561_list

Mathematica

Select[Range[100], AllTrue[{#^2 + 1, 3 # - 1, #^2 + # + 41}, PrimeQ] &] (* Robert Price, Apr 19 2025 *)