A276971 -id:A276971 - 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

A015935 Positive integers n such that 2^n == 2^11 (mod n).

Original entry on oeis.org

1, 2, 3, 4, 8, 11, 14, 15, 16, 31, 32, 35, 51, 56, 64, 121, 128, 146, 224, 256, 341, 451, 455, 496, 508, 512, 671, 781, 896, 1024, 1111, 1235, 1271, 1441, 1547, 1661, 1736, 1991, 2048, 2091, 2101, 2321, 2651, 2761, 2981, 3091, 3421, 3584, 3641, 3731, 3751, 4064, 4088, 4403, 4411, 4631, 4741, 5071, 5401, 5731, 5951, 6171, 6191, 6281, 6386, 6611, 6851, 6941, 7051, 7271, 7601, 7711, 7936, 8261, 8371, 8435, 8456, 8921

Offset: 1

Robert G. Wilson v

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
OEIS Wiki, 2^n mod n

The odd terms form A276971.

m = 2^11; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

isok(n) = Mod(2, n)^n == Mod(2, n)^11; \\ Michel Marcus, Oct 08 2018

Edited by Max Alekseyev, Jul 30 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 *)

A385073 a(n) = b^(n-1) mod n, where b = A053669(n) is the least integer greater than 1 and coprime to n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 3, 4, 3, 1, 5, 1, 3, 4, 11, 1, 11, 1, 7, 4, 3, 1, 5, 16, 3, 13, 27, 1, 7, 1, 11, 4, 3, 9, 29, 1, 3, 4, 27, 1, 17, 1, 27, 31, 3, 1, 29, 15, 33, 4, 27, 1, 11, 49, 3, 4, 3, 1, 43, 1, 3, 4, 43, 16, 23, 1, 27, 4, 13, 1, 29, 1, 3, 34, 27, 9, 5, 1, 27, 40, 3, 1, 17

Offset: 1

Robert G. Wilson v, Jun 16 2025

Inspired by Fermat's Little Theorem.

a(n) > 0 for n > 1 since n and b are coprime.

Wikipedia, Fermat's little theorem

Cf. A053669, A385074.

f:= proc(n) local b;
  b:= 2;
  while n mod b = 0 do b:= nextprime(b) od;
  b &^ (n-1) mod n
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Jun 18 2025

a[n_] := Block[{b = 2}, While[GCD[n, b] > 1, b++]; PowerMod[b, n - 1, n]]; Array[a, 84]

a(n) = forprime(p=2, , if(n%p, return(lift(Mod(p, n)^(n-1))))); \\ Michel Marcus, Jun 18 2025

a(n) = 0 iff n = 1.
a(n) = 1 iff n belongs to A000040, A001567, or A130433.
a(n) = 2 iff n>1 and belongs to A173572;
a(n) = 4 iff n belongs to A033553;
a(n) = 8 iff n>7 and belongs to either A033984 or A173138;
a(n) = 16 iff n>15 and belongs to A276968;
a(n) = 32 iff n>1 and belongs to A215610;
a(n) = 64 iff n>63 and belongs to A276969;
a(n) = 128 iff n>127 and belongs to A215611;
a(n) = 256 iff n>255 and belongs to A276970;
a(n) = 512 iff n>511 and belongs to A215612;
a(n) = 1024 iff n>1023 and belongs to A276971;
a(n) = 2048 iff n>2047 and belongs to A215613;
From Robert Israel, Jun 18 2025: (Start)
a(2*p) = 3 if p is a prime > 3.
a(3*p) = 4 if p is a prime > 2.
a(4*p) = 3^3 if p is a prime > 5.
a(6*p) = 5^5 if p is a prime > 509.
a(8*p) = 3^5 if p is a prime > 271.
a(10*p) = 3^9 if p is a prime > 1951.
a(12*p) = 5^11 if p is a prime > 4069003. (End)

cp's OEIS Frontend

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

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A015935 Positive integers n such that 2^n == 2^11 (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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A276970 Odd integers n such that 2^n == 2^9 (mod n).

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A385073 a(n) = b^(n-1) mod n, where b = A053669(n) is the least integer greater than 1 and coprime to n.

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