A173572 -id:A173572 - cp's OEIS frontend

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

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 *)

A242865 Numbers n such that 3^(n - 3) is congruent to 1 modulo n.

Original entry on oeis.org

3, 9299, 31903, 50963, 87043, 115918, 116891, 219827, 241043, 394243, 550243, 617503, 760243, 806623, 1029253, 1050787, 1458083, 1642798, 1899458, 2864755, 3205387, 3588115, 3839363, 4164578, 5041223, 5610583, 5834755, 5977555, 7837903, 8005558, 8067433, 8128823, 9007603, 9298903, 9449113, 9617443, 9835843

Offset: 1

Felix Fröhlich, May 24 2014

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..372 (terms 1..163 from Felix Fröhlich)

Cf. A067945, A173572.

Intersection with A033553 gives A277344.

Select[Range[10^4], Mod[3^(# - 3), #] == 1 &] (* Alonso del Arte, May 27 2014 *)

for(n=3, 10^6, if(Mod(3, n)^(n-3)==1, print1(n, ", ")))

A192109 Numbers k that divide 2^(k-1) - 2.

Original entry on oeis.org

1, 2, 6, 10, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 170, 178, 182, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526, 538, 542, 554, 562, 566, 586, 614, 622, 626

Offset: 1

Max Alekseyev, Apr 22 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Contains A216090 as subsequence.
Subsequence of A015921, consisting of the terms that are not multiples of 4.
The odd terms form A173572.
Cf. A014741, A000918.

import Data.List (elemIndices)
a192109 n = a192109_list !! (n-1)
a192109_list = map (+ 1) $ elemIndices 0 $ zipWith mod a000918_list [1..]
-- Reinhard Zumkeller, Apr 23 2013

Join[{1,2},Select[Range[700],PowerMod[2,#-1,#]==2&]] (* Harvey P. Dale, May 15 2015 *)

is(n)=Mod(2,n)^(n-1)==2 \\ Charles R Greathouse IV, Nov 04 2016

A346988 a(n) is the smallest k > n such that n^(k-n) == 1 (mod k).

Original entry on oeis.org

2, 20737, 9299, 7, 13, 311, 15, 127, 17, 37, 14, 23, 17, 157, 106, 31, 29, 312953, 45, 95951, 41, 91, 33, 47, 28, 95, 35, 271, 35, 9629, 39, 311, 85, 397, 46, 71, 43, 1793, 95, 79, 61, 821, 51, 18881, 67, 103, 51, 12409, 73, 409969, 65, 87, 65, 71233, 63, 155, 65, 69, 87, 1962251, 91, 2443783, 155

Offset: 1

Thomas Ordowski, Aug 10 2021

nonn

Smallest k > n coprime to n such that n^k == n^n (mod k).

If a(n) is a prime p, then n^(n-1) == 1 (mod p).

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A173572, A242865, A344681.

a[n_] := Module[{k = n + 1}, While[PowerMod[n, k - n, k] != 1, k++]; k]; Array[a, 60] (* Amiram Eldar, Aug 10 2021 *)

a(n) = my(k=n+1); while (Mod(n, k)^(k-n) != 1, k++); k; \\ Michel Marcus, Aug 10 2021

def A346988(n):
    k, kn = n+1, 1
    while True:
        if pow(n,kn,k) == 1:
            return k
        k += 1
        kn += 1 # Chai Wah Wu, Aug 28 2021

More terms from Amiram Eldar, Aug 10 2021

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

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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A242865 Numbers n such that 3^(n - 3) is congruent to 1 modulo n.

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A192109 Numbers k that divide 2^(k-1) - 2.

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A346988 a(n) is the smallest k > n such that n^(k-n) == 1 (mod k).

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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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