A276967 -id:A276967 - cp's OEIS frontend

A015922 Numbers k such that 2^k == 8 (mod k).

1, 2, 3, 4, 8, 9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201, 213, 219, 237, 248, 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

Offset: 1

Robert G. Wilson v

nonn

For all m, 2^A015921(m) - 1 belongs to this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..29055 (first 6822 terms from Zak Seidov)
OEIS Wiki, 2^n mod n.

Contains A033553 as a subsequence.
The odd terms form A276967.
Cf. A015921, A130133, A130134.

a015922Q[n_Integer] := If[Mod[2^n, n] == Mod[8, n], True, False];
a015922[n_Integer] :=
Flatten[Position[Thread[a015922Q[Range[n]]], True]];
a015922[1000000] (* Michael De Vlieger, Jul 16 2014 *)
m = 8; Join[Select[Range[m], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^3], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)
Join[{1,2,3,4,8},Select[Range[650],PowerMod[2,#,#]==8&]] (* Harvey P. Dale, Aug 22 2020 *)

isok(n) = Mod(2, n)^n == Mod(8, n); \\ Michel Marcus, Oct 13 2013, Jul 16 2014

First 5 terms inserted by David W. Wilson

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

Original entry on oeis.org

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

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 *)

A357125 Positive integers n such that 2^(n-3) == -1 (mod n).

Original entry on oeis.org

1, 5, 4553, 46777, 82505, 4290773, 4492205, 4976429, 21537833, 21549349, 51127261, 56786089, 60296573, 80837773, 87761789, 94424465, 138644873, 168865001, 221395541, 255881453, 297460453, 305198249, 360306365, 562654205, 635374253, 673867253, 808333573, 1164757553, 1210317349

Offset: 1

Max Alekseyev, Sep 13 2022

nonn

Also, odd integers n dividing 2^n + 8.

Some large terms: 5603900696716667005, 446661376165868432471569407934747098747181600670953926245, 1533278864164902082788937853692280620552397221686019535813.

Max Alekseyev, Table of n, a(n) for n = 1..781 (all terms below 10^15)

The odd terms of A245319.

Cf. A006521, A115976, A276967, A251603.

Select[Range[2155*10^4],PowerMod[2,#-3,#]==#-1&]//Quiet (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 08 2025 *)

cp's OEIS Frontend

A015922 Numbers k such that 2^k == 8 (mod k).

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

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

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

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A357125 Positive integers n such that 2^(n-3) == -1 (mod n).

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