A015910 -id:A015910 - cp's OEIS frontend

A125000 Integers n such that 2^n == 19 (mod n).

1, 17, 2873, 10081, 3345113, 420048673, 449349533, 2961432773, 19723772249, 821451792317, 1207046362769

Offset: 1

Zak Seidov, Nov 15 2006

No other terms below 10^15. Some larger terms: 500796684074966733196301. - Max Alekseyev, May 23 2012

Cf. A124974, A033981, A050259, A124977, A124965, A015911, A015910.

m = 19; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

Terms 1, 17 prepended by Max Alekseyev, May 20 2011

a(8)-a(11) from Max Alekseyev, May 23 2012

A128126 Numbers k such that 2^k == 18 (mod k).

Original entry on oeis.org

1, 2, 14, 35, 77, 98, 686, 1715, 5957, 18995, 26075, 43921, 49901, 52334, 86555, 102475, 221995, 250355, 1228283, 1493597, 4260059, 6469715, 10538675, 15374219, 19617187, 22731275, 53391779, 60432239, 68597795, 85672139, 175791077

Offset: 1

Alexander Adamchuk, Feb 15 2007

nonn

Joe Crump (joecr(AT)carolina.rr.com) Table of n, a(n) for n = 1..50
Joe K. Crump, 2^n mod n

Cf. A015910, A036236, A050259 (numbers k such that 2^k == 3 (mod k)), A033981, A051447, A033982, A051446, A033983, A128121, A128122, A128123, A128124, A128125.

[1,2,14] cat [n: n in [1..10^8] | Modexp(2, n, n) eq 18]; // Vincenzo Librandi, Apr 05 2019

m = 18; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)
Join[{1,2,14},Select[Range[86*10^6],PowerMod[2,#,#]==18&]] (* Harvey P. Dale, Feb 23 2025 *)

isok(n) = Mod(2, n)^n == 18; \\ Michel Marcus, Oct 09 2018

More terms from Joe Crump (joecr(AT)carolina.rr.com), Mar 04 2007

1, 2 and 14 added by N. J. A. Sloane, Apr 23 2007

A124977 Least positive number k such that 2^k mod k = 2n+1, or 0 if no such k exists.

Original entry on oeis.org

0, 4700063497, 19147, 25, 2228071, 262279, 95, 481, 45, 2873, 3175999, 555, 95921, 174934013, 777, 140039, 2463240427, 477, 91, 623, 2453, 55, 345119, 1131, 943, 21967, 135, 46979, 125, 3811, 23329, 155, 1064959, 245

Offset: 0

Zak Seidov, Nov 14 2006

nonn

			a(3) = 25 because 2^25 = 33554432 = 7 + 25*1342177.

Cf. A015910, A015911, A033981, A036236, A050259, A122182, A124965, A124974.

nk[n_] := Module[ {k}, k = 1;
  While[PowerMod[2, k, k] != 2 n + 1, k++]; k]
Join[{0}, Table[nk[i], {i, 1, 33}]]  (* Robert Price, Oct 11 2018 *)

A bisection of A036236: a(n) = A036236(2n+1).

Edited by Max Alekseyev, May 20 2011

A128125 Numbers k such that 2^k == 14 (mod k).

Original entry on oeis.org

1, 2, 3, 10, 1010, 61610, 469730, 2037190, 3820821, 9227438, 21728810, 24372562, 207034456857, 1957657325241, 2002159320610, 35169368880130, 36496347203230, 116800477091426

Offset: 1

Alexander Adamchuk, Feb 15 2007

No other terms below 10^15. Some larger terms: 279283702428813463, 3075304070192893442, 21894426987819404424310, 4616079845508388554313022889, 82759461944940747300611642693066719359651817521, 446*(2^445-7)/1061319625781480182060453906975 (107 digits). - Max Alekseyev, Oct 03 2016

Joe K. Crump, 2^n mod n

Cf. A015910, A036236, A050259 (numbers k such that 2^k == 3 (mod k)), A033981, A051447, A033982, A051446, A033983, A128121, A128122, A128123, A128124, A128126.

For[n=1, n<= 10^6, n++, If[PowerMod[2,n,n] == Mod[14,n], Print[n]]] (* Stefan Steinerberger, May 05 2007 *)
m = 14; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

1, 2, 3 and 10 added by N. J. A. Sloane, Apr 23 2007
More terms from Stefan Steinerberger, May 05 2007
a(13) from Max Alekseyev, May 15 2011
a(14), a(16), a(17) from Max Alekseyev, Dec 16 2013
a(15), a(18) from Max Alekseyev, Oct 03 2016

A066441 a(n) = 11^n mod n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 4, 1, 8, 1, 0, 1, 11, 9, 11, 1, 11, 1, 11, 1, 8, 11, 11, 1, 1, 17, 26, 25, 11, 1, 11, 1, 11, 19, 16, 1, 11, 7, 5, 1, 11, 1, 11, 33, 26, 29, 11, 1, 18, 1, 5, 29, 11, 1, 11, 9, 20, 5, 11, 1, 11, 59, 8, 1, 46, 55, 11, 21, 20, 11, 11, 1, 11, 47, 26, 49, 44, 25, 11, 1

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n: A015910 (k=2), A066601 (k=3), A066602 (k=4), A066603 (k=5), A066604 (k=6), A066438 (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), this sequence (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(11^n,n),n=1..80); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[11, n, n], {n, 80} ]
```

a(n) = { lift(Mod(11, n)^n) } \\ Harry J. Smith, Feb 14 2010

A124965 Odd values of 2^n mod n corresponding to the n's given in A015911.

Original entry on oeis.org

7, 17, 43, 37, 13, 17, 57, 53, 85, 53, 63, 151, 93, 161, 107, 173, 67, 193, 251, 239, 43, 233, 107, 155, 161, 105, 81, 179, 233, 103, 239, 143, 125, 179, 349, 161, 305, 89, 257, 83, 143, 279, 197, 161, 371, 35, 15, 449, 253, 437, 403, 407, 255, 279, 353

Offset: 1

Zak Seidov, Nov 14 2006, corrected Nov 25 2006

nonn

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A015911.

Select[PowerMod[2,#,#]&/@Range[550],OddQ] (*Ray Chandler, Nov 20 2011*)

a(n) = A015910(A015911(n)).

Edited by Ray Chandler at the suggestion of Jon E. Schoenfield, Nov 20 2011

A064367 a(n) = 2^n mod prime(n).

Original entry on oeis.org

0, 1, 3, 2, 10, 12, 9, 9, 6, 9, 2, 26, 33, 1, 9, 28, 33, 27, 13, 48, 8, 36, 47, 4, 95, 20, 76, 62, 23, 4, 8, 117, 68, 25, 138, 64, 150, 43, 61, 10, 72, 156, 40, 12, 73, 51, 48, 41, 24, 26, 71, 48, 32, 16, 128, 173, 74, 110, 118, 59, 30, 247, 202, 208, 284, 53, 128, 32, 139

Offset: 1

Labos Elemer, Sep 27 2001

nonn

Below the exponent n=10000, some integers (like 5,7,14,17,19,22,...,44, etc.) are not yet present among residues. Will they appear later?
For a(n) with n <= 10^6, the following residues have not yet appeared: {19, 22, 46, 52, 57, 65, 70, 77, 81, 85, 88, 90, 91, 103, 104, 106, 108, 115, 120, 122, 123, 125, ..., 15472319} (14537148 terms). - Michael De Vlieger, Jul 16 2017
Heuristically, the probability of 2^n mod prime(n) taking a given value is approximately 1/prime(n) for large n. Since the sum of 1/prime(n) diverges, we should expect each positive integer to appear infinitely many times in the sequence. However, since the sum diverges very slowly, the first n where it appears may be very large. - Robert Israel, Jul 17 2017

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000079, A015910.

seq(2 &^ n mod ithprime(n), n=1..100); # Robert Israel, Jul 17 2017

Array[PowerMod[2, #, Prime@ #] &, 69] (* Michael De Vlieger, Jul 16 2017 *)

a(n) = { lift(Mod(2,prime(n))^n) } \\ Harry J. Smith, Sep 12 2009

a(n) = A000079(n) mod A000040(n).

Definition corrected by Harry J. Smith, Sep 12 2009

A073798 pi(n) is a power of 2, where pi(n) = A000720(n) is the number of primes <= n.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 19, 20, 21, 22, 53, 54, 55, 56, 57, 58, 131, 132, 133, 134, 135, 136, 311, 312, 719, 720, 721, 722, 723, 724, 725, 726, 1619, 1620, 3671, 3672, 8161, 8162, 8163, 8164, 8165, 8166, 17863, 17864, 17865, 17866, 17867, 17868, 17869, 17870

Offset: 1

Labos Elemer, Aug 14 2002

nonn

The numbers occur in blocks of consecutive integers: 2, 3-4, 7-10, 19-22, ...; the n-th block starts at the 2^n-th prime (A033844) and ends just before the (2^n + 1)-th prime (A051439).

			10 is in the sequence since pi(10)=4=2^2.

Chai Wah Wu, Table of n, a(n) for n = 1..1103 (n = 1..665 from Ivan Neretin)

Cf. A000079, A000720, A015910, A073797, A073799.

pow2[n_] := n==1||(n>1&&IntegerQ[n/2]&&pow2[n/2]); Select[Range[20000], pow2[PrimePi[ # ]]&]
Flatten@Table[Range[p = Prime[2^k], NextPrime[p] - 1], {k, 0, 11}] (* Ivan Neretin, Jan 21 2017 *)

isok(n) = my(pi = primepi(n)); (pi==1) || (pi==2) || (ispower(primepi(n),,&k) && (k==2)); \\ Michel Marcus, Jan 23 2017

Edited by Dean Hickerson, Aug 15 2002

A096196 a(n) = (1 + 2^n) mod n.

Original entry on oeis.org

0, 1, 0, 1, 3, 5, 3, 1, 0, 5, 3, 5, 3, 5, 9, 1, 3, 11, 3, 17, 9, 5, 3, 17, 8, 5, 0, 17, 3, 5, 3, 1, 9, 5, 19, 29, 3, 5, 9, 17, 3, 23, 3, 17, 18, 5, 3, 17, 31, 25, 9, 17, 3, 29, 44, 33, 9, 5, 3, 17, 3, 5, 9, 1, 33, 65, 3, 17, 9, 45, 3, 65, 3, 5, 69, 17, 19, 65, 3, 17, 0, 5, 3, 65, 33, 5, 9, 81, 3

Offset: 1

Labos Elemer, Jul 26 2004

Michael De Vlieger, Table of n, a(n) for n = 1..10000
OEIS Wiki, 2^n mod n

Cf. A006521, A015910, A082495.

[(1+2^n) mod (n): n in [1..100]]; // Vincenzo Librandi, Dec 11 2015

seq(1 + 2 &^ n mod n, n = 1 .. 250); # Robert Israel, Dec 10 2015

Table[Mod[1 + Mod[2, n]^n, n], {n, 89}] (* Michael De Vlieger, Dec 10 2015 *)

a(n)=(1+2^n)%n \\ Anders Hellström, Dec 10 2015

a(n)=lift(1+Mod(2,n)^n); \\ Michel Marcus, Dec 12 2015

A106262 An invertible triangle of remainders of 2^n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 2, 1, 0, 1, 0, 4, 2, 1, 0, 2, 0, 3, 4, 2, 1, 0, 1, 0, 1, 2, 4, 2, 1, 0, 2, 0, 2, 4, 1, 4, 2, 1, 0, 1, 0, 4, 2, 2, 0, 4, 2, 1, 0, 2, 0, 3, 4, 4, 0, 8, 4, 2, 1, 0, 1, 0, 1, 2, 1, 0, 7, 8, 4, 2, 1, 0, 2, 0, 2, 4, 2, 0, 5, 6, 8, 4, 2, 1, 0, 1, 0, 4, 2, 4, 0, 1, 2, 5, 8, 4, 2, 1

Offset: 0

Paul Barry, Apr 28 2005

			Triangle begins:
  1;
  0, 1;
  0, 2, 1;
  0, 1, 2, 1;
  0, 2, 0, 2, 1;
  0, 1, 0, 4, 2, 1;
  0, 2, 0, 3, 4, 2, 1;
  0, 1, 0, 1, 2, 4, 2, 1;
  0, 2, 0, 2, 4, 1, 4, 2, 1;
  0, 1, 0, 4, 2, 2, 0, 4, 2, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A106263 (row sums), A106264 (diagonal sums).

Cf. A000034, A015910, A062173, A112983, A213859, A294389, A294390.

[Modexp(2, n-k, k+2): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 10 2023

Table[PowerMod[2, n-k, k+2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 10 2023 *)

flatten([[power_mod(2,n-k,k+2) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 10 2023

T(n, k) = 2^(n-k) mod (k+2).
Sum_{k=0..n} T(n, k) = A106263(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A106264(n) (diagonal sums).
From G. C. Greubel, Jan 10 2023: (Start)
T(n, 0) = A000007(n).
T(n, 1) = A000034(n+1).
T(2*n, n) = A213859(n).
T(2*n, n-1) = A015910(n+1).
T(2*n, n+1) = A294390(n+3).
T(2*n+1, n-1) = A112983(n+1).
T(2*n+1, n+1) = A294389(n+3).
T(2*n-1, n-1) = A062173(n+1). (End)

cp's OEIS Frontend

A125000 Integers n such that 2^n == 19 (mod n).

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A128126 Numbers k such that 2^k == 18 (mod k).

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Magma

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A124977 Least positive number k such that 2^k mod k = 2n+1, or 0 if no such k exists.

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Mathematica

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A128125 Numbers k such that 2^k == 14 (mod k).

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A066441 a(n) = 11^n mod n.

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Maple

Mathematica

PARI

A124965 Odd values of 2^n mod n corresponding to the n's given in A015911.

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Mathematica

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Extensions

A064367 a(n) = 2^n mod prime(n).

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Maple

Mathematica

PARI

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Extensions

A073798 pi(n) is a power of 2, where pi(n) = A000720(n) is the number of primes <= n.

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