Primes of the form Phi_n(2)/(k*n+1)
 

Phi_n(2) is prime for these n:

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, 579, 590, 600, 607, 626, 690, 694, 712, 745, 795, 816, 897, 909, 954, 990, 1106, 1192, 1224, 1230, 1279, 1384, 1386, 1402, 1464, 1512, 1554, 1562, 1600, 1670, 1683, 1727, 1781, 1834, 1904, 1990, 1992, 2008, 2037, 2203, 2281, 2298, 2353, 2406, 2456, 2499, 2536, 2838, 3006, 3074, 3217, 3415, 3418, 3481, 3766, 3817, 3927, ...

Now, let's search primes of the form Phi_n(2)/(k*n+1) for fixed k (since all prime factors > n of Phi_n(2) are = 1 mod n, thus they are of the form k*n+1)

Phi_n(2)/(n+1) is prime for these n:

18, 28, 36, 58, 60, 66, 82, 106, 138, 1290, 3010, 6658, ...

Phi_n(2)/(2*n+1) is prime for these n:

11, 20, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, ...

Phi_n(2)/(3*n+1) is prime for these n:

36, 94, 214, 1086, 5894, ...

Phi_n(2)/(4*n+1) is prime for these n:

28, 70, 88, 144, 1470, 7830, ...

Phi_n(2)/(6*n+1) is prime for these n:

37, 72, 121, 153, 221, 245, 688, ...

Phi_n(2)/(8*n+1) is prime for these n:

11, 1121, 2544, 3320, 4205, ...

If we change the number 2 to 10, then they are related to the repunit numbers.

Phi_n(10) is prime for these n:

2, 4, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, ...

Phi_n(10)/(n+1) is prime for these n:

6, 16, 18, 58, 130, 592, 618, 658, 708, 1326, ...

Phi_n(10)/(2*n+1) is prime for these n:

1, 6, 15, 33, 98, 440, ...

Phi_n(10)/(3*n+1) is prime for these n:

230, ...

Phi_n(10)/(4*n+1) is prime for these n:

(none found)

If we change the number 2 to 12, then there are related to the dozenal (duodecimal) repunits:

Phi_n(12) is prime for these n:

1, 2, 3, 5, 10, 12, 19, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511, 636, 675, 701, 793, 945, 1090, 1268, 1272, 1971, ...

Phi_n(12)/(n+1) is prime for these n:

4, 6, 256, ...

Phi_n(12)/(2*n+1) is prime for these n:

36, 95, 120, 131, 1044, ...

Phi_n(12)/(3*n+1) is prime for these n:

6, 154, ...

Phi_n(12)/(4*n+1) is prime for these n:

9, 324, ...