A187787 -id:A187787 - cp's OEIS frontend

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

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

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

A081856 Numbers k such that 2k-1 divides 2^k-1.

Original entry on oeis.org

1, 2, 8, 128, 228, 648, 1352, 1908, 3240, 4608, 5220, 5976, 11448, 13160, 13920, 21528, 22050, 23760, 23940, 24840, 30960, 31284, 31584, 31968, 32768, 37224, 46092, 46512, 47268, 60480, 65664, 66528, 78540, 78600, 81728, 82800, 84312, 98406, 102672, 103968

Offset: 1

Benoit Cloitre, Apr 11 2003

nonn

Subsequence of odd terms is given by A233415. - Charles R Greathouse IV, Dec 04 2013

Numbers 2k-1 form a subsequence of A187787. - Max Alekseyev, Sep 04 2024

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A187787, A233415.

a:= proc(n) option remember; local k;
      if n=1 then 1 else for k from 1+a(n-1)
      while 2&^k mod(2*k-1)<>1 do od; k fi
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, May 27 2016

terms = 100; Reap[For[n=1; k=1, k <= terms, n++, If[Divisible[2^n-1, 2n-1], Print[k, " ", n]; Sow[n]; k++]]][[2, 1]] (* Jean-François Alcover, Apr 06 2017 *)
Join[{1},Select[Range[110000],PowerMod[2,#,2*#-1]==1&]] (* Harvey P. Dale, Jan 19 2019 *)

is(n)=Mod(2,2*n-1)^n==1 \\ Charles R Greathouse IV, Dec 04 2013

a(38)-a(40) from Michel Marcus, Dec 04 2013

A233415 Odd numbers k such that 2k-1 divides 2^k-1.

Original entry on oeis.org

1, 763425, 10888425, 40068105, 142086921, 191345625, 462784725, 468545025, 552451809, 595018305, 683993905, 956917125, 1013987349, 1024992045, 1567781325, 1581567885, 3094868865, 3312888345, 4839991545, 4882263477, 5064476505, 5613455925, 7303900125

Offset: 1

Charles R Greathouse IV, Dec 09 2013

nonn

These seem to be much rarer than the corresponding even numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 1..51

Subsequence of A081856.

Numbers 2a(n)-1 form a subsequence of A187787.

PARI
```
is(n)=n%2 && Mod(2, 2*n-1)^n==1
```

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.

A069927 Numbers k that divide 2^(k+3) - 1.

Original entry on oeis.org

1, 3, 5, 9, 21, 45, 63, 85, 105, 117, 273, 285, 357, 585, 627, 765, 1365, 1397, 1449, 1677, 1989, 2337, 3597, 3705, 3885, 4221, 4365, 4485, 4797, 4953, 5061, 5607, 5797, 7137, 7521, 7565, 7665, 8109, 10197, 10545, 10845, 11445, 11565, 12597, 13065, 13717, 14637

Offset: 1

Benoit Cloitre, Apr 25 2002

			k = 1 divides 2^(k+3) - 1 = 15, so 1 is a term.

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 500 terms from Harvey P. Dale)

Cf. A187787.

q:= k-> 0=(2&^(k+3)-1) mod k:
select(q, [$1..20000])[];  # Alois P. Heinz, Apr 10 2025

Select[Range[15000],PowerMod[2,#+3,#]==1&] (* Harvey P. Dale, Jul 09 2018 *)

Missing term 1 inserted by Alois P. Heinz, Apr 10 2025

A307217 Semiprimes pq such that 2^(p+q) == 1 (mod pq).

Original entry on oeis.org

9, 15, 35, 119, 5543, 74447, 90859, 110767, 222179, 389993, 1526849, 2927297, 3626699, 4559939, 24017531, 137051711, 160832099, 229731743, 627699239, 880021141, 1001124539, 1041287603, 1104903617, 1592658611, 1717999139, 8843679683, 15575602979, 15614760199, 20374337479

Offset: 1

Thomas Ordowski, Mar 29 2019

nonn

For k > 9, these are semiprimes k such that 2^(k+1) == 1 (mod k): semiprimes in A187787.

In this sequence, only 9 is a perfect square. - Jinyuan Wang, Mar 30 2019

Cf. A001358, A046315, A127104, A187787, A208728.

isok(k) = (bigomega(k)==2) && (Mod(2, k)^(k+1) == 1); \\ (for k > 9) Michel Marcus, Mar 29 2019

use ntheory ":all"; forsemiprimes { print "$\n" if powmod(2, vecsum(factor($)), $) == 1 } 4, 1e7; # _Daniel Suteu, Mar 30 2019

a(7)-a(18) from Amiram Eldar, Mar 29 2019

a(19)-a(29) from Daniel Suteu, Mar 29 2019

A319538 Numbers k such that k divides 2^(2*k+1) - 1.

Original entry on oeis.org

1, 7, 217, 1057, 3937, 10447, 24601, 32767, 91657, 145337, 279527, 666967, 1412113, 2484247, 2874847, 3124327, 4169137, 4472167, 9526207, 12021439, 16539337, 16646017, 19384207, 20139367, 24639727, 28127137, 28940887, 30583087, 66131279, 68068777, 70694167, 72299857, 72903847, 73498471, 87507049

Offset: 1

Altug Alkan, Sep 22 2018

nonn

Sequence is infinite because 2^A187787(t) - 1 is a term for all t >= 1.

Cf. A187787, A296369, A319222.

Select[Range[10^5], Mod[2^(2 # + 1) - 1, #] == 0 &] (* Michael De Vlieger, Sep 24 2018 *)

PARI
```
isok(n) = Mod(2, n)^(2*n+1)==1;
```

A334634 Numbers m that divide 2^m + 11.

Original entry on oeis.org

1, 13, 16043199041, 91118493923, 28047837698634913

Offset: 1

Max Alekseyev, Sep 10 2020

Equivalently, numbers m such that 2^m == -11 (mod m).

No other terms below 10^17.

OEIS Wiki, 2^n mod n

Solutions to 2^n == k (mod n): A296370 (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), this sequence (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).

a(5) from Sergey Paramonov, Oct 10 2021

A381010 Positive integers k such that 2^(k+2) - 1 is divisible by k.

Original entry on oeis.org

1, 7, 511, 713, 11023, 15553, 43873, 81079, 95263, 323593, 628153, 2275183, 6520633, 6955513, 7947583, 10817233, 12627943, 14223823, 15346303, 19852423, 27923663, 28529473, 29360527, 31019623, 39041863, 41007823, 79015273, 134217727, 143998193, 213444943, 227018383

Offset: 1

Oisín Flynn-Connolly, Apr 10 2025

nonn

7 is the only prime term.

Cf. A000225, A055685, A069927, A187787.

q:= k-> 0=(2&^(k+2)-1) mod k:
select(q, [$1..1000000])[];  # Alois P. Heinz, Apr 10 2025

isok(k) = Mod(2, k)^(k+2) == 1; \\ Michel Marcus, Apr 10 2025

def in_sequence(n):
    return pow(2, n + 2, n) == 1 % n

cp's OEIS Frontend

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

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

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A081856 Numbers k such that 2k-1 divides 2^k-1.

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A233415 Odd numbers k such that 2k-1 divides 2^k-1.

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

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A069927 Numbers k that divide 2^(k+3) - 1.

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A307217 Semiprimes p*q such that 2^(p+q) == 1 (mod p*q).

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A319538 Numbers k such that k divides 2^(2*k+1) - 1.

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A307217 Semiprimes pq such that 2^(p+q) == 1 (mod pq).