A015910 -id:A015910 - cp's OEIS frontend

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

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

A056969 a(n) = 10^n modulo n.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 3, 0, 1, 0, 10, 4, 10, 2, 10, 0, 10, 10, 10, 0, 13, 12, 10, 16, 0, 22, 1, 4, 10, 10, 10, 0, 10, 32, 5, 28, 10, 24, 25, 0, 10, 22, 10, 12, 10, 8, 10, 16, 31, 0, 31, 16, 10, 28, 10, 16, 31, 42, 10, 40, 10, 38, 55, 0, 30, 34, 10, 4, 34, 60, 10, 64, 10, 26, 25, 44, 54, 40

Offset: 1

Henry Bottomley, Jul 20 2000

nonn

			a(7) = 3 since 10000000 = 7*1428571+3

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. k^n mod n: A015910 (k=2), A066601 (k=3), A066602 (k=4), A066603 (k=5), A066604 (k=6), A066438 (k=7), A066439 (k=8), A066440 (k=9), this sequence (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(10^n,n),n=1..78); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[10, n, n], {n, 80} ]
```

a(n) = lift(Mod(10, n)^n); \\ Michel Marcus, Oct 19 2017

a(n) = 10*A056968(n) mod n = A011557(n) mod n.

A066438 a(n) = 7^n mod n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 0, 1, 1, 9, 7, 1, 7, 7, 13, 1, 7, 1, 7, 1, 7, 5, 7, 1, 7, 23, 1, 21, 7, 19, 7, 1, 13, 15, 28, 1, 7, 11, 31, 1, 7, 7, 7, 25, 37, 3, 7, 1, 0, 49, 37, 9, 7, 1, 43, 49, 1, 49, 7, 1, 7, 49, 28, 1, 37, 37, 7, 21, 67, 49, 7, 1, 7, 49, 43, 45, 28, 25, 7, 1

Offset: 1

Robert G. Wilson v, Dec 27 2001

nonn

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n; A015910 (k=2), A066601 (k=3), A066602 (k=4), A066603 (k=5), A066604 (k=6), this sequence (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(7^n,n),n=1..80); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[7, n, n], {n, 80} ]
```

a(n) = { lift(Mod(7, n)^n) } \\ Harry J. Smith, Feb 14 2010

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

A082495 a(n) = (2^n - 1) mod n.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 7, 7, 3, 1, 3, 1, 3, 7, 15, 1, 9, 1, 15, 7, 3, 1, 15, 6, 3, 25, 15, 1, 3, 1, 31, 7, 3, 17, 27, 1, 3, 7, 15, 1, 21, 1, 15, 16, 3, 1, 15, 29, 23, 7, 15, 1, 27, 42, 31, 7, 3, 1, 15, 1, 3, 7, 63, 31, 63, 1, 15, 7, 43, 1, 63, 1, 3, 67, 15, 17, 63, 1, 15, 79, 3, 1, 63, 31, 3, 7, 79

Offset: 1

Anonymous, Apr 28 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A015910, A048298.

Cf. A000079, A062173.

a082495 n = a015910 n + a048298 n - 1  -- Reinhard Zumkeller, Oct 17 2015

Table[Mod[2^m-1,m],{m,6!}] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2010 *)

vector(80, n, (2^n-1) %n) \\ Michel Marcus, Jan 16 2015

def A082495(n): return (m if (m:=pow(2,n,n)) else n)-1 # Chai Wah Wu, Dec 01 2022

a(n) = A015910(n) + A048298(n) - 1.

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

A066603 a(n) = 5^n mod n.

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 5, 1, 8, 5, 5, 1, 5, 11, 5, 1, 5, 1, 5, 5, 20, 3, 5, 1, 0, 25, 26, 9, 5, 25, 5, 1, 26, 25, 10, 1, 5, 25, 8, 25, 5, 1, 5, 9, 35, 25, 5, 1, 19, 25, 23, 1, 5, 1, 45, 25, 11, 25, 5, 25, 5, 25, 62, 1, 5, 49, 5, 13, 56, 65, 5, 1, 5, 25, 50, 17, 3, 25, 5, 65, 80, 25, 5, 1, 65

Offset: 1

Amarnath Murthy, Dec 22 2001

			a(7) = 5 as 5^7 = 78125 = 7*11160 + 5.

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. k^n mod n: A015910 (k=2), A066601 (k=3), A066602 (k=4), this sequence (k=5), A066604 (k=6), A066438 (k=7), A066439 (k=8), A066440 (k=9), A056969 (k=10), A066441 (k=11), A066442 (k=12), A116609 (k=13).

seq(irem(5^n,n),n=1..85); # Zerinvary Lajos, Apr 20 2008

Mathematica
```
Table[PowerMod[5, n, n], {n, 85} ]
```

a(n) = { lift(Mod(5, n)^n) } \\ Harry J. Smith, Mar 09 2010

More terms from Robert G. Wilson v, Dec 27 2001

cp's OEIS Frontend

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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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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A056969 a(n) = 10^n modulo n.

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A066438 a(n) = 7^n mod n.

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Maple

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PARI

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

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Mathematica

Extensions

A082495 a(n) = (2^n - 1) mod n.

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