A215613 -id:A215613 - cp's OEIS frontend

A176997 Integers k such that 2^(k-1) == 1 (mod k).

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

Offset: 1

Juri-Stepan Gerasimov, Dec 08 2010

nonn

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
The union of {1}, the odd primes, and the Fermat pseudoprimes, i.e., {1} U A065091 U A001567. Also, the union of A006005 and A001567 (conjectured by Alois P. Heinz, Dec 10 2010). - Max Alekseyev, Sep 22 2016
These numbers were called "fermatians" by Shanks (1962). - Amiram Eldar, Apr 21 2024

			5 is in the sequence because 2^(5-1) == 4^(5-1) == 8^(5-1) == 1 (mod 5).

Daniel Shanks, Solved and Unsolved Problems in Number Theory, Spartan Books, Washington D.C., 1962.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015919.
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).
Cf. A000079, A062173, A062175, A176817.

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 *)

isok(n) = Mod(2, n)^(n-1) == 1; \\ Michel Marcus, Sep 23 2016

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
A176997_list = list(islice(A176997_gen(),30)) # Chai Wah Wu, Jun 30 2022

Edited by Max Alekseyev, Sep 22 2016

A215610 Odd integers n such that 2^n == 2^6 (mod n).

Original entry on oeis.org

1, 31, 18631, 55831, 92329, 3014633, 3556559, 6429121, 9664591, 12158831, 33554431, 34844431, 566740481, 644903881, 727815241, 842608801, 2207017049, 2208171881, 2445644207, 8694918511, 9031128791, 18738146881, 27345981361, 35476604081

Offset: 1

Max Alekseyev, Aug 17 2012

nonn

Also, the odd solutions to 2^(n-6) == 1 (mod n). The only even solution is n=6.

For all m, 2^A033981(m)-1 belongs to this sequence.

Max Alekseyev, Table of n, a(n) for n = 1..102 (all terms below 10^14)

The odd terms of A015926.

Cf. A173572, A033984, A215611, A215612, A215613

m = 64; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

A215611 Odd integers n such that 2^n == 2^8 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Aug 17 2012

nonn

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.

Max Alekseyev, Table of n, a(n) for n = 1..859 (all terms below 10^14)

The odd terms of A015929.

Cf. A051447, A033984, A173572, A215610, A215612, A215613.

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 *)

A215612 Odd integers n such that 2^n == 2^10 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Aug 17 2012

nonn

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.

Max Alekseyev, Table of n, a(n) for n = 1..209 (all terms below 10^14)

The odd terms of A015932.

Cf. A173572, A033984, A215610, A215611, A215613

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 *)

A015937 Positive integers n such that 2^n == 2^12 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 23, 24, 32, 36, 40, 42, 48, 60, 62, 64, 68, 72, 80, 84, 89, 96, 120, 126, 128, 132, 144, 156, 160, 168, 180, 192, 204, 228, 240, 252, 256, 276, 288, 312, 320, 336, 340, 348, 352, 360, 372, 384, 420, 444, 462

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215613.

For all m, 2^A051446(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015929, A015931, A015932, A015935.

With[{c=2^12},Select[Range[1,6000],Divisible[2^#-c,#]&]] (* Harvey P. Dale, Mar 20 2011 *)

Edited by Max Alekseyev, Jul 31 2011

A276967 Odd integers n such that 2^n == 2^3 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Sep 22 2016

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.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015922.

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).

Join[{1}, Select[Range[1, 1023, 2], PowerMod[2, # - 3, #] == 1 &]] (* Alonso del Arte, Sep 22 2016 *)

isok(n) = (n % 2) && (Mod(2,n)^n==8); \\ Michel Marcus, Sep 23 2016

A276968 Odd integers n such that 2^n == 2^5 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n-5) == 1 (mod n).
Contains A050993 as a subsequence.
For all m, 2^A128122(m)-1 belongs to this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015925.
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).
Cf. A050993, A128122.

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 *)

A276969 Odd integers n such that 2^n == 2^7 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n-7) == 1 (mod n).
Contains A208155 as a subsequence.
For all m, 2^A015922(m)-1 belongs to this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015927.
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).
Cf. A015922, A208155.

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 *)

is(n)=n%2 && Mod(2,n)^n==128 \\ Charles R Greathouse IV, Sep 22 2016

A276970 Odd integers n such that 2^n == 2^9 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n-9) == 1 (mod n).
Contains A208157 as a subsequence.
For all m, 2^A128123(m)-1 belongs to this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015931.

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).

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 *)

A276971 Odd integers n such that 2^n == 2^11 (mod n).

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n-11) == 1 (mod n).

For all m, 2^A128124(m)-1 belongs to this sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015935.
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).
Cf. A128124.

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 *)

cp's OEIS Frontend

A176997 Integers k such that 2^(k-1) == 1 (mod k).

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A215610 Odd integers n such that 2^n == 2^6 (mod n).

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A215611 Odd integers n such that 2^n == 2^8 (mod n).

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A215612 Odd integers n such that 2^n == 2^10 (mod n).

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A015937 Positive integers n such that 2^n == 2^12 (mod n).

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A276967 Odd integers n such that 2^n == 2^3 (mod n).

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A276968 Odd integers n such that 2^n == 2^5 (mod n).

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A276969 Odd integers n such that 2^n == 2^7 (mod n).

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