A128121 -id:A128121 - cp's OEIS frontend

A033984 Odd integers n such that 2^n == 16 (mod n).

1, 7, 40369, 673663, 990409, 1697609, 2073127, 6462649, 7527199, 7559479, 14421169, 21484129, 37825753, 57233047, 130647919, 141735559, 179203369, 188967289, 218206489, 259195009, 264538057, 277628449, 330662479, 398321239, 501126487

Offset: 1

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

nonn

The odd terms of A015924.

For all m, 2^A128121(m)-1 belongs to this sequence.

Max Alekseyev, Table of n, a(n) for n = 1..790 (all terms below 10^14)

Besides initial terms, the sequence coincides with A173138.

Cf. A015924, A173138, A173572.

Select[Range[1,510000001,2],PowerMod[2,#,#]==16&] (* Harvey P. Dale, Dec 11 2010 *)

Edited and terms 1,7 prepended by Max Alekseyev, Aug 09 2012

A116611 Positive integers n such that 13^n == 5 (mod n).

Original entry on oeis.org

1, 2, 4, 44, 82, 236, 25433, 177764, 219244, 86150213, 107218402, 1260236441, 12856300141, 447650116364, 657175627369, 14543842704596, 125035120614917

Offset: 1

Zak Seidov, Feb 19 2006

nonn

No other terms below 10^15.

Some larger terms: 99790373907467602, 846248577183963835642742, 273781047810302314432122404459324, 4174626353309446327489382394518975030641698849116, 211*(13^211-5)/12607932861823674049268705845744 (207 digits). - Max Alekseyev, Jun 29 2011

			44 is in this sequence because 13^44 = 10315908977942302627204470186314316211062255002161 = 234452476771415968800101595143507186615051250049*44 + 5 == 5 (mod 44).

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), this sequence (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609, A128121, A276740, A123091.

Join[{1, 2}, Select[Range[1000000], PowerMod[13, #, #] == 5 &]] (* Robert Price, Apr 10 2020 *)

is(n) = Mod(13,n)^n==5; \\ Charles R Greathouse IV, Jun 08 2015

More terms from Ryan Propper, Apr 01 2006

Terms 1,2,4 are prepended and a(13)-a(17) are added by Max Alekseyev, Jun 29 2011, Nov 27 2017

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

A015924 Positive integers n such that 2^n == 16 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 12, 16, 20, 24, 28, 40, 44, 48, 52, 60, 68, 76, 80, 92, 112, 116, 120, 124, 148, 154, 164, 172, 188, 204, 208, 212, 236, 240, 244, 264, 268, 280, 284, 292, 316, 332, 340, 356, 364, 388, 404, 412, 428, 436, 452, 508, 520, 524, 548, 556, 596

Offset: 1

Robert G. Wilson v

nonn

Odd terms are given by A033984.

For all m, 2^A128121(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 A050992 as a subsequence.

Cf. A033984, A173138.

Select[Range[1000], Mod[2^# - 2^4, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)
Join[{1,2,4,6,7,8,12,16},Select[Range[600],PowerMod[2,#,#]==16&]] (* Harvey P. Dale, Dec 03 2021 *)

Edited and terms 1,2,4,6,7,8,12,16 prepended by Max Alekseyev, Jul 29 2011

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

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

A245318 Numbers k that divide 2^k + 5.

Original entry on oeis.org

1, 7, 133, 1517, 11761, 676333, 1484413, 3627557, 10289371, 1449045241, 2433687407, 12309023183, 29013950411, 11701492535299, 223598572318157, 362232879754103

Offset: 1

Derek Orr, Jul 17 2014

No other terms below 10^15. Some large terms: 37367159696063084325121, 1637537600494693555095121, 50692913747901869910332539, 407*(2^407+5)/1125038874668278099 (108 digits). - Max Alekseyev, Sep 22 2016

			2^7 + 5 = 133 is divisible by 7. Thus 7 is a term of this sequence.

OEIS Wiki, 2^n mod n

Cf. A128121, A168614.

Select[Range[10^5], Divisible[2^# + 5, #] &] (* Robert Price, Oct 12 2018 *)

for(n=1,10^9,if(Mod(2,n)^n==Mod(-5,n),print1(n,", ")))

a(10)-a(13) from Lars Blomberg, Nov 05 2014

a(14)-a(16) from Max Alekseyev, Oct 09 2016

cp's OEIS Frontend

A033984 Odd integers n such that 2^n == 16 (mod n).

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A116611 Positive integers n such that 13^n == 5 (mod n).

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

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A015924 Positive integers n such that 2^n == 16 (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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A128126 Numbers k such that 2^k == 18 (mod k).

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