A050259 -id:A050259 - cp's OEIS frontend

A015910 a(n) = 2^n mod n.

0, 0, 2, 0, 2, 4, 2, 0, 8, 4, 2, 4, 2, 4, 8, 0, 2, 10, 2, 16, 8, 4, 2, 16, 7, 4, 26, 16, 2, 4, 2, 0, 8, 4, 18, 28, 2, 4, 8, 16, 2, 22, 2, 16, 17, 4, 2, 16, 30, 24, 8, 16, 2, 28, 43, 32, 8, 4, 2, 16, 2, 4, 8, 0, 32, 64, 2, 16, 8, 44, 2, 64, 2, 4, 68, 16, 18, 64, 2, 16, 80, 4, 2, 64

Offset: 1

Robert G. Wilson v

nonn

2^n == 2 mod n if and only if n is a prime or a member of A001567 or of A006935. [Guy]. - N. J. A. Sloane, Mar 22 2012; corrected by Thomas Ordowski, Mar 26 2016
Known solutions to 2^n == 3 (mod n) are given in A050259.
This sequence is conjectured to include every integer k >= 0 except k = 1. A036236 includes a proof that k = 1 is not in this sequence, and n = A036236(k) solves a(n) = k for all other 0 <= k <= 1000. - David W. Wilson, Oct 11 2011
It could be argued that a(0) := 1 would make sense, e.g., thinking of "mod n" as "in Z/nZ", and/or because (anything)^0 = 1. See also A112987. - M. F. Hasler, Nov 09 2018

			a(7) = 2 because 2^7 = 128 = 2 mod 7.
a(8) = 0 because 2^8 = 256 = 0 mod 8.
a(9) = 8 because 2^9 = 512 = 8 mod 9.

Richard K. Guy, Unsolved Problems in Number Theory, F10.

T. D. Noe, Table of n, a(n) for n=1..10000
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 4
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.
Peter L. Montgomery, 65-digit solution.

Cf. A036236, A015911, A015921, A001567.

Cf. A000079, A062173, A082495.

import Math.NumberTheory.Moduli (powerMod)
a015910 n = powerMod 2 n n  -- Reinhard Zumkeller, Oct 17 2015

[Modexp(2, n, n): n in [1..100]]; // Vincenzo Librandi, Nov 09 2018

A015910 := n-> modp(2 &^ n,n) ; # Zerinvary Lajos, Feb 15 2008

Mathematica
```
Table[PowerMod[2, n, n], {n, 85}]
```

makelist(power_mod(2, n, n), n, 1, 84); /* Bruno Berselli, May 20 2011 */

a(n)=lift(Mod(2,n)^n) \\ Charles R Greathouse IV, Jul 15 2011

a(2^k) = 0. - Alonso del Arte, Nov 10 2014

a(n) == 2^(n-phi(n)) mod n, where phi(n) = A000010(n). - Thomas Ordowski, Mar 26 2016

A015921 Positive integers n such that 2^n == 4 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 132, 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, 372, 382, 386, 394, 398, 422, 446

Offset: 1

Robert G. Wilson v

nonn

Odd terms are given by A173572.

For all m, 2^A050259(m)-1 belongs to this sequence.

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

Contains A050990 as a subsequence.

Cf. A015910, A173572.

Select[Range[500], PowerMod[2, #, #] == 4 &] (* Alonso del Arte, Jul 07 2011 *)

Edited and terms 1,2,4 prepended by Max Alekseyev, Jul 29 2011

A015973 Positive integers n such that n | (3^n + 2).

Original entry on oeis.org

1, 5, 77, 278377, 3697489, 219596687717, 56865169816619

Offset: 1

Robert G. Wilson v

No other terms below 10^15. Some larger term: 3142423971953435020522506484187. - Max Alekseyev, Aug 04 2011

Cf. A015940, A050259, A123062

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), this sequence (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277126 (k=7), A277274 (k=11).

a(1)=1 prepended and a(6)-a(7) added by Max Alekseyev, Aug 04 2011

A173572 Odd integers n such that 2^n == 4 (mod n).

Original entry on oeis.org

1, 20737, 93527, 228727, 373457, 540857, 2231327, 11232137, 15088847, 15235703, 24601943, 43092527, 49891487, 66171767, 71429177, 137134727, 207426737, 209402327, 269165561, 302357057, 383696711, 513013327

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

The odd terms of A015921.
Also, nonprime integers n such that 2^(n-2) == 1 (mod n).
For all m, 2^A050259(m)-1 belongs to this sequence.
If n > 1 is a term and p is a primitive prime factor of 2^(n-2)-1, then n*p is also a term. Hence, the sequence is infinite. (Rotkiewicz 1984)

A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). (in Russian)
R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, Third Edition, 2004
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

Max Alekseyev, Table of n, a(n) for n = 1..722 (all terms below 10^14)
C. K. Caldwell, Composite Numbers
A. Rotkiewicz, On the congruence 2^(n-2) == 1 (mod n). Math. Comp. 43 (1984), 271-272.

Cf. A015921, A050259, A173138, A002808.

with(numtheory): for n from 1 to 100000000 do: a:= 2^(n-2)- 1; b:= a / n; c:= floor(b): if b = c and tau(n) <> 2 then print (n); else fi;od:

m = 4; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^6, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)

is(n) = n%2==1 && Mod(2,n)^n==Mod(4,n) \\ Jinyuan Wang, Feb 22 2019

Edited and term 1 prepended by Max Alekseyev, Aug 09 2012

A116629 Positive integers k such that 13^k == 3 (mod k).

Original entry on oeis.org

1, 2, 5, 166, 287603, 9241538, 2366680105, 8347156585, 21682897793, 6988245760865, 9045859950329, 10076294257985, 50299408064905, 254874726648713

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 24 2017

Some larger terms: 1440926367749746685, 76025040962646716305439353859479569558065. - Max Alekseyev, Jun 29 2011

Cf. A116609, A050259, A122780, A130422, A123061, A277554.

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), this sequence (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620 (k=10), A116638 (k=11), A116639 (k=15).

Join[{1, 2}, Select[Range[1, 5000], Mod[13^#, #] == 3 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1, 2}, Select[Range[10000000], PowerMod[13, #, #] == 3 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 3; \\ Michel Marcus, Nov 19 2017

Two more terms from Ryan Propper, Jan 09 2008

Terms 1,2 are prepended and a(9)-a(14) are added by Max Alekseyev, Jun 29 2011; Nov 24 2017

A033982 Integers n such that 2^n == 11 (mod n).

Original entry on oeis.org

1, 3, 262279, 143823239, 382114303, 1223853491, 381541784791, 556985326431, 6236258437049, 98828020264153

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

894157816841269897394424491194255510200782699 belongs to this sequence. [From Max Alekseyev]

Joe K. Crump, 2^n mod n

Cf. A125000, A124974, A033983, A033981, A050259, A124977, A124965, A015911, A015910.

Cf. A015910, A015911, A033981, A050259, A124965, A124974, A124977, A015910, A015911, A033983, A125000.

m = 11; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^3], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar
Terms 1, 3 prepended by Max Alekseyev, May 18 2011
a(9), a(10) from Max Alekseyev, Jul 30 2011

A276671 Positive integers k such that 3^k == 2 (mod k).

Original entry on oeis.org

1, 2929, 9742277641, 23341869101, 15092205901438895, 16311037042239935

Offset: 1

Max Alekseyev, Oct 05 2016

No other terms below 2*10^16. A larger term: 31744873758348589012852097851.

Cf. A015973, A067945, A015949, A055686, A242865, A234535, A234536, A050259.

Join[{1}, Select[Range[10000], PowerMod[3, #, #] == 2 &]] (* Alonso del Arte, Oct 11 2016 *)

isok(n) = Mod(3, n)^n == Mod(2, n); \\ Dmitry Ezhov, Sep 28 2016

Order of terms corrected by Felix Fröhlich, Oct 06 2016

a(5)-a(6) from Sergey Paramonov, Oct 03 2021

A033983 Integers n such that 2^n == 15 (mod n).

Original entry on oeis.org

1, 13, 481, 44669, 1237231339, 1546675117, 62823773963, 284876771881, 1119485807557, 26598440989093

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

No other terms below 10^14.

Joe K. Crump, 2^n mod n
OEIS Wiki, 2^n mod n

Cf. A125000, A124974, A033982, A033981, A050259, A124977, A124965, A015911, A015910.

Cf. A015910, A015911, A033981, A050259, A124965, A124974, A124977, A015910, A015911, A033982, A125000.

m = 15; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^3], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

One more term from Joe K. Crump (joecr(AT)carolina.rr.com), Jun 20 2000
Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar
Terms 1, 13 prepended by Max Alekseyev, May 18 2011
a(10) from Max Alekseyev, Dec 15 2013

A051447 Integers n such that 2^n == 9 (mod n).

Original entry on oeis.org

1, 7, 2228071, 16888457, 352978207, 1737848873, 77362855777, 567442642711

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

No other terms below 10^15. [Max Alekseyev, May 20 2012]

Joe K. Crump, 2^n mod n

Cf. A125000, A124974, A033983, A033982, A033981, A050259, A124977, A124965, A015911, A015910.

Cf. A033981, A033982, A033983.

m = 9; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^3], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

Edited by N. J. A. Sloane, Jun 22 2008, at the suggestion of Don Reble

Terms 1, 7 prepended by Max Alekseyev, May 18 2011

A128121 Numbers k such that 2^k == 5 (mod k).

Original entry on oeis.org

1, 3, 19147, 129505699483, 674344345281, 1643434407157, 5675297754009, 12174063716147, 162466075477787, 313255455573801, 324082741109271

Offset: 1

Alexander Adamchuk, Feb 15 2007

Joe K. Crump, 2^n mod n
OEIS Wiki, 2^n mod n

Cf. A015910, A036236, A050259 (numbers k such that 2^k == 3 (mod k)), A033981, A051447, A033982, A051446, A033983, A128122, A128123, A128124, A128125, A128126.

m = 5; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

1 and 3 added by N. J. A. Sloane, Apr 23 2007

Missing a(10) inserted by Sergey Paramonov, Sep 06 2021

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A015910 a(n) = 2^n mod n.

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A015921 Positive integers n such that 2^n == 4 (mod n).

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A015973 Positive integers n such that n | (3^n + 2).

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A173572 Odd integers n such that 2^n == 4 (mod n).

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A116629 Positive integers k such that 13^k == 3 (mod k).

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A033982 Integers n such that 2^n == 11 (mod n).

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A276671 Positive integers k such that 3^k == 2 (mod k).

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A033983 Integers n such that 2^n == 15 (mod n).

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