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COMMENTS
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All nonzero a(n) are equal to powers of 2. a((p-1)/2) = 1 for prime p>2. - Alexander Adamchuk, Sep 17 2006
Further comments from Alexander Adamchuk, Oct 01 2006: (Start)
a(19) is unknown.
a(20)-a(24) = {1,1,16,1,4}.
a(26)-a(30) = {1,2,2,1,1}.
a(32) = -1.
a(33) = 1.
a(35)-a(42) = {1,1,2,16,1,4,1,2}.
a(44)-a(45) = {1,2}.
a(47)-a(48) = {2,1}.
a(50)-a(51) = {1,1}.
a(53)-a(60) = {1,1,2,1,32,2,4,2}.
a(62)-a(63) = {2,1}.
a(64) = -1.
a(65)-a(71) = {1,4,2,1,1,4,4}.
a(73)-a(83) = {2,1,1,8,4,1,16,2,1,4,1}.
a(85)-a(90) = {2,1,4,2,1,1}.
a(92) = 2.
a(94)-a(99) = {16,1,1,4,1,1}.
a(102)-a(105) = {2,2,8,1}.
a(n) is unknown for n = {19,25,31,34,43,46,49,52,61,72,84,91,93,100,101,...}.
Corresponding smallest primes of the form (2n)^k +1 are listed in A084712[n] = {3,5,7,0,11,13,197,17,19,401,23,577,677,29,31,0,1336337,37,...} Smallest prime of the form (2n)^k +1, or 0 if no such number exists.
The first occurrence of a(k) = 2^n is k = {1,7,17,76,22,57,137,117,307,...} = A122528[n] Minimum number k such that (2k)^(2^n) + 1 is prime, or A079706[A122528(n)] = 2^n.
Corresponding primes A084712[A122528(n)] = {3,197,1336337,284936905588473857,197352587024076973231046657,...}. (End)
The terms a(32) and a(64) are known to be -1 because 2^(6k)+1 and 2^(7k)+1 are divisible by 4^k+1 and 2^k+1, respectively, for all k >0. Also, a(45)=2 because 8101 is prime. - T. D. Noe, May 13 2008
a(19) >= 2^20 or a(19) = -1. - Robert Price, Mar 02 2015
From Robert G. Wilson v, Aug 30 2016: (Start)
n = 1: 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, ..., ;
n = 2: 7, 10, 12, 13, 27, 28, 37, 42, 45, 47, 55, 58, 60, 62, 67, 73, 80, 85, 88, 92, 102, 103, 112, 115, 118, 130, 132, 142, 150, 157, 163, 170, 175, 192, 193, 203, 218, 220, 222, 232, 235, 237, 248, 268, 272, 292, 297, 317, 318, 322, ..., ;
n = 4: 17, 24, 40, 59, 66, 70, 71, 77, 82, 87, 97, 110, 121, 124, 127, 133, 136, 139, 144, 148, 160, 164, 167, 182, 187, 207, 236, 238, 244, 246, 247, 252, 258, 263, 275, 277, 283, 291, 312, 314, 328, 351, 355, 365, 374, 379, 389, 394, ..., ;
n = 8: 76, 104, 145, 196, 213, 217, 255, 271, 298, 305, 332, 391, 433, 442, 446, 458, 467, 478, 511, 514, 560, 612, 616, 628, 642, 655, 695, 801, 814, 841, 934, 968, 1039, 1045, 1050, 1097, 1137, 1141, 1164, 1181, 1189, 1240, 1245, ..., ;
n = 16: 22, 38, 79, 94, 159, 185, 226, 280, 344, 368, 387, 388, 395, 415, 416, 417, 423, 450, 486, 492, 522, 539, 607, 706, 764, 867, 906, 917, 928, 945, 992, 1036, 1078, 1104, 1109, 1115, 1142, 1159, 1176, 1224, 1231, 1281, 1456, 1631, ..., ;
n = 32: 57, 166, 171, 180, 188, 214, 294, 402, 425, 599, 839, 857, 909, 980, 1059, 1209, 1217, 1334, 1387, 1393, 1422, 1434, 1521, 1522, 1599, 1744, 1748, 1757, 1904, 2217, 2245, 2250, 2266, 2458, 2467, 2532, 541, 2579, 2606, 2610, ..., ;
n = 64: 137, 206, 364, 542, 815, 902, 1082, 1247, 1262, 1307, 1392, 1512, 1639, 1814, 1847, 1875, 2015, 2029, 2083, 2162, 2359, 2607, 2859, 2947, 3218, 3346, 3421, 3456, 3481, 3542, 3566, 4065, 4261, 4416, 4494, 4496, 4570, 4720, ..., ;
n = 128: 117, 253, 266, 274, 1738, 2894, 3040, 3375, 3846, 4853, 5119, 5497, 5716, 5777, 6850, 7007, 7144, 7783, 8485, 8980, 96965, ..., ;
n = 256: 307, 449, 674, 787, 969, 1061, 1139, 1381, 1717, 2047, 2102, 2856, 2872, 4322, 4381, 4404, 4571, 5446, 6103, 6610, 6611, 6685, 6869, 7057, 7963, 8128, 8358, 9671, ..., ;
n = 512: 671, 1577, 1939, 2232, 2344, 2687, 2849, 2885, 3193, 3341, 3694, 4339, 4396, 5734, 6377, 6599, 6888, 7367, 8413, 9457, ..., ;
n = 1024: 412, 738, 816, 1242, 1532, 3320, 4535, 6274, 6497, 8823, 9168, 9782, ..., ;
n = 2048: 1279, 6618, 7524,..., ;
n = 4098: 767, 8791, 9112,..., ;
n = 8192: 35926,..., ;
n = 16384: 50915,..., ;
n = 32768: 35453,..., ;
n = 65536: 24297,..., ; etc.
(End)
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