A015911 -id:A015911 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

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

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

A124974 Integers n such that 2^n == 17 (mod n).

Original entry on oeis.org

1, 3, 5, 9, 45, 99, 53559, 1143357, 2027985, 36806085, 1773607905, 3314574181, 1045463125509, 1226640523999, 3567404505159, 28726885591099, 39880799734039, 87977068273719, 106436400721299, 339966033494859, 999567363539883

Offset: 1

Zak Seidov, Nov 14 2006

nonn

Some larger terms: 576541379659648320485

			2^45 = 17 + 45*781874935307,
2^99 = 17 + 99*6402275758728431320690420229.

Cf. A033981, A124965, A015911.

m = 17; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

Terms 1, 3, 5, 9 prepended by Max Alekseyev, May 20 2011

a(11)-a(21) from Max Alekseyev, May 25 2012

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

Original entry on oeis.org

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

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

A226221 Numbers n such that 2^n mod n is not a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 25, 27, 32, 35, 36, 42, 45, 49, 50, 54, 55, 64, 70, 75, 77, 81, 88, 91, 95, 98, 99, 100, 104, 105, 108, 110, 115, 117, 119, 121, 125, 128, 130, 135, 136, 140, 143, 147, 150, 152, 153, 155, 156, 160, 161, 162, 169, 171, 175, 180, 184, 187, 189, 190, 198, 200

Offset: 1

J. M. Bergot and Charles R Greathouse IV, May 31 2013

nonn

All terms beyond the first two are composite: this is a subsequence of A065090.

			2^18 = 262144 = 10 mod 18 and 10 is not a power of 2, so 18 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A015910, A015911.

isA226221 := proc(n)
    local m ;
    if n <= 2 then
        return true;
    end if;
    m := A015910(n) ;
    if type(m,'odd') or m = 0 then
        true;
    elif nops(numtheory[factorset](m))  >1 then
        true;
    else
        false;
    end if;
end proc:
A226221 := proc(n)
    local a;
    if n <= 2 then
        n;
    else
        for a from procname(n-1)+1 do
            if isA226221(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A226221(n),n=1..30) ; # R. J. Mathar, Jun 06 2013

Select[Range[200],!IntegerQ[Log[2,PowerMod[2,#,#]]]&] (* Harvey P. Dale, Feb 28 2022 *)

PARI
```
ispow2(n)=n>0 && n==1<
				
```

A056740 Odd numbers k such that 2^k mod k = 2^(k+2) mod (k-2) is also an odd number.

Original entry on oeis.org

135, 1423249, 31491395, 55519333, 1065373685, 3559609381

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com), Aug 25 2000

			135 is a term because 2^135 mod 135 = 53 and 2^137 mod 133 = 53.

Joe K. Crump, 2^n mod n

Cf. A015911.

isok(n) = {va = Mod(2,n)^n; (lift(va) % 2) && (lift(va) == lift(Mod(2, n-2)^(n+2)));} \\ Michel Marcus, Sep 02 2013

a(6) and title clarified by Sean A. Irvine, May 04 2022

cp's OEIS Frontend

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

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

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

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

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

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A124974 Integers n such that 2^n == 17 (mod n).

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

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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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