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Integers k such that 2^(k-1) == 1 (mod k).
+10
13
1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331
COMMENTS
Old definition was: Odd integers n such that 2^(n-1) == 4^(n-1) == 8^(n-1) == ... == k^(n-1) (mod n), where k = A062383(n). Dividing 2^(n-1) == 4^(n-1) (mod n) by 2^(n-1), we get 1 == 2^(n-1) (mod n), implying the current definition. - Max Alekseyev, Sep 22 2016
These numbers were called "fermatians" by Shanks (1962). - Amiram Eldar, Apr 21 2024
REFERENCES
Daniel Shanks, Solved and Unsolved Problems in Number Theory, Spartan Books, Washington D.C., 1962.
EXAMPLE
5 is in the sequence because 2^(5-1) == 4^(5-1) == 8^(5-1) == 1 (mod 5).
MATHEMATICA
m = 1; Join[Select[Range[m], Divisible[2^(# - 1) - m, #] &],
Select[Range[m + 1, 10^3], PowerMod[2, # - 1, #] == m &]] (* Robert Price, Oct 12 2018 *)
PROG
(PARI) isok(n) = Mod(2, n)^(n-1) == 1; \\ Michel Marcus, Sep 23 2016
(Python)
from itertools import count, islice
def A176997_gen(startvalue=1): # generator of terms >= startvalue
if startvalue <= 1:
yield 1
k = 1<<(s:=max(startvalue, 1))-1
for n in count(s):
if k % n == 1:
yield n
k <<= 1
CROSSREFS
Odd integers n such that 2^n == 2^k (mod n): this sequence (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), A276970 (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).
Odd integers n such that 2^n == 2^8 (mod n).
+10
11
1, 127, 3473, 19313, 30353, 226703, 230777, 345023, 929783, 1790159, 1878073, 2569337, 3441743, 4213511, 8026103, 9770153, 19139183, 24261623, 30652223, 35482433, 38044223, 40642103, 55015793, 65046479, 67411121, 69601193, 119611073
COMMENTS
Also, the odd solutions to 2^(n-8) == 1 (mod n). The only even solution is n=8.
For all m, 2^ A051447(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^8; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)
Odd integers n such that 2^n == 2^10 (mod n).
+10
11
1, 7, 73, 9271, 3195367, 6769801, 15413863, 24540337, 47424961, 52268743, 146583343, 384586849, 469501471, 475882081, 859764727, 1097475991, 1169323417, 1400034919, 2518532047, 2870143993, 3258854623, 5609707729, 6022970047, 6420870271, 9011348521
COMMENTS
Also, the odd solutions to 2^(n-10) == 1 (mod n). The only even solution is n=10.
For all m, 2^ A033982(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^10; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^7, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 15 2018 *)
Odd integers n such that 2^n == 2^12 (mod n).
+10
11
1, 23, 89, 2047, 5969, 1803407, 6531977, 6667079, 7694041, 16252897, 146825647, 284464633, 315096487, 351745417, 413414167, 512694047, 615366953, 2723423687, 3104303327, 3969298807, 5754671737, 7242954137, 8766711119, 14046374879
COMMENTS
Also, the odd solutions to 2^(n-12) == 1 (mod n). The only even solution is n=12.
For all m, 2^ A051446(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^12; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^7, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 15 2018 *)
Positive integers n such that 2^n == 2^6 (mod n).
+10
9
1, 2, 4, 6, 8, 10, 12, 16, 18, 24, 30, 31, 32, 36, 42, 48, 64, 66, 72, 78, 84, 90, 96, 102, 114, 126, 138, 144, 168, 174, 176, 186, 192, 210, 222, 234, 246, 252, 258, 282, 288, 318, 336, 354, 366, 390, 396, 402, 426, 438, 456, 474, 496, 498, 504, 510, 534, 546
COMMENTS
The odd terms are given by A215610.
For all m, 2^ A033981(m)-1 belongs to this sequence.
MATHEMATICA
Select[Range[1000], Mod[2^# - 2^6, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)
CROSSREFS
Cf. A015919, A015921, A015922, A015924, A015925, A015927, A015929, A015931, A015932, A015935, A015937.
Odd integers n such that 2^n == 2^3 (mod n).
+10
8
1, 3, 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201, 213, 219, 237, 249, 267, 291, 303, 309, 315, 321, 327, 339, 381, 393, 399, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 693, 699, 717, 723, 731, 753, 771, 789, 807
COMMENTS
Also, integers n such that 2^(n - 3) == 1 (mod n).
Contains A033553 as a subsequence. Smallest term greater than 3 missing in A033553 is 731.
For all m, 2^ A015921(m) - 1 belongs to this sequence.
MATHEMATICA
Join[{1}, Select[Range[1, 1023, 2], PowerMod[2, # - 3, #] == 1 &]] (* Alonso del Arte, Sep 22 2016 *)
PROG
(PARI) isok(n) = (n % 2) && (Mod(2, n)^n==8); \\ Michel Marcus, Sep 23 2016
CROSSREFS
Odd integers n such that 2^n == 2^k (mod n): A176997 (k = 1), A173572 (k = 2), this sequence (k = 3), A033984 (k = 4), A276968 (k = 5), A215610 (k = 6), A276969 (k = 7), A215611 (k = 8), A276970 (k = 9), A215612 (k = 10), A276971 (k = 11), A215613 (k = 12).
Odd integers n such that 2^n == 2^5 (mod n).
+10
7
1, 3, 5, 25, 65, 85, 145, 165, 185, 205, 221, 265, 305, 365, 445, 465, 485, 505, 545, 565, 685, 745, 785, 825, 865, 905, 965, 985, 1025, 1085, 1145, 1165, 1205, 1285, 1345, 1385, 1405, 1465, 1565, 1585, 1685, 1705, 1745, 1765, 1865, 1925, 1945, 1985, 2005, 2045, 2105, 2165, 2245, 2285, 2305, 2325
COMMENTS
Also, integers n such that 2^(n-5) == 1 (mod n).
For all m, 2^ A128122(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^5; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^3, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)
CROSSREFS
Odd integers n such that 2^n == 2^k (mod n): A176997 (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), this sequence (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), A276970 (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).
Odd integers n such that 2^n == 2^7 (mod n).
+10
7
1, 3, 7, 15, 49, 91, 133, 217, 255, 259, 301, 427, 469, 511, 527, 553, 679, 721, 763, 889, 973, 1015, 1057, 1099, 1141, 1267, 1351, 1393, 1477, 1561, 1603, 1687, 1897, 1939, 1981, 2107, 2149, 2191, 2317, 2359, 2443, 2569, 2611, 2653, 2779, 2863, 2947, 3031, 3073, 3199, 3241, 3409, 3493, 3661, 3787
COMMENTS
Also, integers n such that 2^(n-7) == 1 (mod n).
For all m, 2^ A015922(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^7; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^3, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)
CROSSREFS
Odd integers n such that 2^n == 2^k (mod n): A176997 (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), this sequence (k=7), A215611 (k=8), A276970 (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).
Odd integers n such that 2^n == 2^9 (mod n).
+10
7
1, 3, 5, 9, 17, 21, 27, 45, 63, 99, 105, 117, 153, 171, 189, 207, 261, 273, 279, 333, 369, 387, 423, 429, 477, 513, 531, 549, 585, 603, 639, 657, 711, 747, 801, 873, 909, 927, 945, 963, 981, 1017, 1143, 1179, 1197, 1209, 1233, 1251, 1341, 1359, 1365, 1413, 1467, 1503, 1557, 1611, 1629, 1665, 1719, 1737
COMMENTS
Also, integers n such that 2^(n-9) == 1 (mod n).
For all m, 2^ A128123(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^9; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^3, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 15 2018 *)
CROSSREFS
Odd integers n such that 2^n == 2^k (mod n): A176997 (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), this sequence (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).
Odd integers n such that 2^n == 2^11 (mod n).
+10
7
1, 3, 11, 15, 31, 35, 51, 121, 341, 451, 455, 671, 781, 1111, 1235, 1271, 1441, 1547, 1661, 1991, 2091, 2101, 2321, 2651, 2761, 2981, 3091, 3421, 3641, 3731, 3751, 4403, 4411, 4631, 4741, 5071, 5401, 5731, 5951, 6171, 6191, 6281, 6611, 6851, 6941, 7051, 7271, 7601, 7711, 8261, 8371, 8435, 8921
COMMENTS
Also, integers n such that 2^(n-11) == 1 (mod n).
For all m, 2^ A128124(m)-1 belongs to this sequence.
MATHEMATICA
m = 2^11; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)
CROSSREFS
Odd integers n such that 2^n == 2^k (mod n): A176997 (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), A276970 (k=9), A215612 (k=10), this sequence (k=11), A215613 (k=12).
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