A051446 -id:A051446 - cp's OEIS frontend

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

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

A128122 Numbers m such that 2^m == 6 (mod m).

Original entry on oeis.org

1, 2, 10669, 6611474, 43070220513807782

Offset: 1

Alexander Adamchuk, Feb 15 2007

No other terms below 10^17. - Max Alekseyev, Nov 18 2022

A large term: 862*(2^861-3)/281437921287063162726198552345362315020202285185118249390789 (203 digits). - Max Alekseyev, Sep 24 2016

			2 == 6 (mod 1), so 1 is a term;
4 == 6 (mod 2), so 2 is a term.

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

Solutions to 2^m == k (mod m): A000079 (k=0),A187787 (k=1/2), A296369 (k=-1/2), A006521 (k=-1), A296370 (k=3/2), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), this sequence (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Cf. A015910, A036236.

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

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

a(5) from Max Alekseyev, Nov 18 2022

A215613 Odd integers n such that 2^n == 2^12 (mod n).

Original entry on oeis.org

1, 23, 89, 2047, 5969, 1803407, 6531977, 6667079, 7694041, 16252897, 146825647, 284464633, 315096487, 351745417, 413414167, 512694047, 615366953, 2723423687, 3104303327, 3969298807, 5754671737, 7242954137, 8766711119, 14046374879

Offset: 1

Max Alekseyev, Aug 17 2012

nonn

Also, the odd solutions to 2^(n-12) == 1 (mod n). The only even solution is n=12.

For all m, 2^A051446(m)-1 belongs to this sequence.

Max Alekseyev, Table of n, a(n) for n = 1..93 (all terms below 10^14)

The odd terms of A015937.

Cf. A173572, A033984, A215610, A215611, A215612

m = 2^12; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^7, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 15 2018 *)

A128123 Numbers k such that 2^k == 10 (mod k).

Original entry on oeis.org

1, 2, 6, 18, 16666, 262134, 4048124214, 24430928839, 243293052886, 41293676570106, 3935632929857549

Offset: 1

Alexander Adamchuk, Feb 15 2007

Some larger terms: 266895924489780149, 2335291686841914329, 18494453435532853111

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, A128124, A128125, A128126.

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

1, 2 and 6 added by N. J. A. Sloane, Apr 23 2007
Missing terms a(9)-a(10) added by Max Alekseyev, Dec 16 2013
a(11) from Max Alekseyev, Sep 27 2016

A296369 Numbers m such that 2^m == -1/2 (mod m).

Original entry on oeis.org

1, 5, 65, 377, 1189, 1469, 25805, 58589, 134945, 137345, 170585, 272609, 285389, 420209, 538733, 592409, 618449, 680705, 778805, 1163065, 1520441, 1700945, 2099201, 2831009, 4020029, 4174169, 4516109, 5059889, 5215769

Offset: 1

Max Alekseyev, Dec 10 2017

nonn

Equivalently, 2^(m+1) == -1 (mod m), or m divides 2^(m+1) + 1.

The sequence is infinite, see A055685.

Chai Wah Wu, Table of n, a(n) for n = 1..1192
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A296370 (k=3/2), A187787 (k=1/2), this sequence (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Select[Range[10^5], Divisible[2^(# + 1) + 1, #] &] (* Robert Price, Oct 11 2018 *)

A296369_list = [n for n in range(1,10**6) if pow(2,n+1,n) == n-1] # Chai Wah Wu, Nov 04 2019

a(n) = A055685(n) - 1.

Incorrect term 4285389 removed by Chai Wah Wu, Nov 04 2019

A128124 Numbers k such that 2^k == 12 (mod k).

Original entry on oeis.org

1, 2, 4, 5, 3763, 125714, 167716, 1803962, 2895548, 4031785, 36226466, 16207566916, 103742264732, 29000474325364, 51053256144532, 219291270961199, 1611547934753332, 5816826177630619

Offset: 1

Alexander Adamchuk, Feb 15 2007

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

Cf. A015910, A036236, A050259, A033981, A051447, A033982, A051446, A033983, A128121, A128122, A128123, A128125, A128126.

m = 12; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

More terms from Ryan Propper, Mar 23 2007
1, 2, 4 and 5 added by N. J. A. Sloane, Apr 23 2007
a(13)-a(15) from Max Alekseyev, May 19 2011
a(15) corrected, a(16)-a(18) added by Max Alekseyev, Oct 02 2016

A015937 Positive integers n such that 2^n == 2^12 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 23, 24, 32, 36, 40, 42, 48, 60, 62, 64, 68, 72, 80, 84, 89, 96, 120, 126, 128, 132, 144, 156, 160, 168, 180, 192, 204, 228, 240, 252, 256, 276, 288, 312, 320, 336, 340, 348, 352, 360, 372, 384, 420, 444, 462

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215613.

For all m, 2^A051446(m)-1 belongs to this sequence.

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

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015929, A015931, A015932, A015935.

With[{c=2^12},Select[Range[1,6000],Divisible[2^#-c,#]&]] (* Harvey P. Dale, Mar 20 2011 *)

Edited by Max Alekseyev, Jul 31 2011

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

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

A296370 Numbers m such that 2^m == 3/2 (mod m).

Original entry on oeis.org

1, 111481, 465793, 79036177, 1781269903307, 250369632905747, 708229497085909, 15673900819204067

Offset: 1

Max Alekseyev, Dec 11 2017

Equivalently, 2^(m+1) == 3 (mod m).
Also, numbers m such that 2^(m+1) - 2 is a Fermat pseudoprime base 2, i.e., 2^(m+1) - 2 belongs to A015919 and A006935.
Some larger terms (may be not in order): 2338990834231272653581, 341569682872976768698011746141903924998969680637.

OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): this sequence (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Select[Range[10^6], Divisible[2^(# + 1) - 3, #] &] (* Robert Price, Oct 11 2018 *)

a(n) = A296104(n) - 1.

cp's OEIS Frontend

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

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A128122 Numbers m such that 2^m == 6 (mod m).

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A215613 Odd integers n such that 2^n == 2^12 (mod n).

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A128123 Numbers k such that 2^k == 10 (mod k).

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A296369 Numbers m such that 2^m == -1/2 (mod m).

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A128124 Numbers k such that 2^k == 12 (mod k).

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A015937 Positive integers n such that 2^n == 2^12 (mod n).

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

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