A036236 -id:A036236 - cp's OEIS frontend

A178194 Smallest k such that 33^k mod k = n.

1, 2, 31, 5, 29, 7, 21, 13, 13684967, 10, 23, 14, 15, 538, 19, 42, 17, 35, 25, 49, 16861, 60, 55, 26, 1157, 38, 511, 54, 30197665, 106, 14691, 46, 155, 37, 18791, 62, 369, 164, 145, 93, 63517, 92, 115, 1046, 3113077, 58, 1376107, 1042, 105, 50, 221

Offset: 0

Artur Jasinski, May 22 2010

nonn

smallest k such that m^k mod k = n
m=2 see A036236
m=3 see A078457
m=4 see A119678
m=5 see A119679
m=6 see A127816
m=7 see A119715
m=8 see A119714
m=9 see A127817
m=10 see A127818
m=11 see A127819
m=12 see A127820
m=13 see A127821
m=14 see A128154
m=15 see A128155
m=16 see A128156
m=17 see A128157
m=18 see A128158
m=19 see A128159
m=20 see A128160
m=21 see A128361
m=22 see A128362
m=23 see A128363
m=24 see A128364
m=25 see A128365
m=26 see A128366
m=27 see A128367
m=28 see A128368
m=29 see A128369
m=30 see A128370
m=31 see A128371
m=32 see A128372
m=33 see A178194
m=34 see A178195
m=35 see A178196
m=36 see A178197
m=37 see A178198
m=38 see A178199
m=39 see A178200
m=40 see A178201
m=41 see A178202

see comment line.

aa = {}; Do[k = 1; While[PowerMod[33, k, k] != n, k++ ]; Print[k]; AppendTo[aa, k], {n, 0, 50}]; aa

A178202 Smallest k such that 41^k mod k = n.

Original entry on oeis.org

1, 2, 3, 19, 37, 76, 7, 17, 9, 22, 31, 15, 29, 77, 309, 34, 7194589, 26, 23, 341, 21, 55, 799, 1658, 476983, 46, 27, 427, 629, 52, 142241, 138, 68889, 136, 1897, 129, 30935, 44, 19303, 1642, 34943, 43, 8858994648397, 102, 117, 436, 7715, 86, 49

Offset: 0

Artur Jasinski, May 23 2010

nonn

Cf. A036236, A078457, A119678, A119679, A119714-A119716, A127817-A127821, A128154-A128372, A178194-A178202

aa = {}; Do[k = 1; While[PowerMod[41, k, k] != n, k++ ]; Print[{n, k}]; AppendTo[aa, k], {n, 1, 50}]; aa

a(0)=1 prepended and a(42) added by Max Alekseyev, Feb 04 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

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

A128370 a(n) = least k such that the remainder of 30^k divided by k is n.

Original entry on oeis.org

29, 7, 26997, 13, 8471, 33, 23, 11, 721, 55, 19, 39, 17, 886, 21, 26, 803, 98, 13289, 22, 51, 878, 1141, 146, 35, 38, 111, 218, 515267673651961, 31, 3212679202339, 56, 267, 866, 4367, 42, 10129, 862, 57, 86, 42691, 13479, 949, 214, 95, 77, 7633, 52, 1469, 170, 429, 68, 2791229, 94, 215, 422, 3849, 842, 9773, 140

Offset: 1

Alexander Adamchuk, Feb 27 2007

nonn

Robert G. Wilson v et al., Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A129369, A128371, A128372.
Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160.
Cf. A128149, A128150, A128172.

t = Table[0, {10000} ]; k = 1; While[ k < 5100000000, a = PowerMod[30, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

Terms a(29) onward from Max Alekseyev, Mar 22 2012

A129492 Composite numbers k such that 2^k mod k is a power of 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 38, 39, 40, 44, 46, 48, 51, 52, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 72, 74, 76, 78, 80, 82, 84, 85, 86, 87, 90, 92, 93, 94, 96, 102, 106, 111, 112, 114, 116, 118, 120, 122, 123, 124, 126, 129, 132, 133, 134, 138

Offset: 1

Robert G. Wilson v, Apr 17 2007

Complement to composite numbers: 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, ....

			15 is a term since 2^15 mod 15 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A036236, A129493, A129494, A129495, A129496, A129497.

Contains A001567.

[k:k in [2..150]| not IsPrime(k)  and  not IsZero(a)  and (PrimeDivisors(a) eq [2]) where a is 2^k mod k ]; // Marius A. Burtea, Dec 04 2019

filter:= proc(n) local k;
  if isprime(n) then return false fi;
  k:= 2 &^ n mod n;
  k > 1 and k = 2^padic:-ordp(k,2)
end proc:
select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

Select[Range@ 141, IntegerQ@ Log[2, PowerMod[2, #, # ]] &]

A129493 Composite numbers k such that 3^k mod k is a power of 3.

Original entry on oeis.org

6, 10, 12, 14, 18, 22, 24, 26, 30, 33, 34, 36, 38, 39, 46, 51, 54, 56, 57, 58, 62, 63, 66, 69, 72, 74, 78, 82, 86, 87, 90, 91, 92, 93, 94, 99, 104, 106, 108, 111, 112, 116, 117, 118, 120, 121, 122, 123, 124, 129, 132, 134, 135, 141, 142, 144, 146, 148, 154, 158, 159

Offset: 1

Robert G. Wilson v, Apr 17 2007

Complement to composite numbers: 9, 15, 21, 25, 27, 28, 35, 42, 44, 45, 48, 49, 50, 52, 55, 60, 65, 68, 70, 75, ....

			14 is a member of the sequence since 3^14 mod 14 = 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A036236, A122780, A129492, A129494, A129495, A129496, A129497.

[k:k in [2..160]| not IsPrime(k)  and  not IsZero(a)  and (PrimeDivisors(a) eq [3]) where a is 3^k mod k ]; // Marius A. Burtea, Dec 04 2019

filter:= proc(n) local k;
  if isprime(n) then return false fi;
  k:= 3 &^ n mod n;
  k > 1 and k = 3^padic:-ordp(k,3)
end proc:
select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

Select[Range@ 161, IntegerQ@ Log[3, PowerMod[3, #, # ]] &]

A129494 Composite numbers k such that 4^k mod k is a power of 4 greater than 1.

Original entry on oeis.org

6, 12, 15, 20, 22, 24, 26, 28, 30, 34, 38, 40, 46, 48, 56, 58, 60, 62, 66, 69, 72, 74, 77, 80, 82, 84, 85, 86, 87, 88, 91, 93, 94, 96, 102, 104, 105, 106, 111, 117, 118, 120, 122, 123, 126, 129, 132, 134, 140, 141, 142, 144, 146, 158, 159, 166, 168, 170, 177, 178, 182

Offset: 1

Robert G. Wilson v, Apr 17 2007

Complement to composite numbers: 4, 8, 9, 10, 14, 16, 18, 21, 25, 27, 32, 33, 35, 36, 39, 42, 44, 45, 49, 50, 51, 52, 54, 55, 57, ... - R. J. Mathar, May 16 2008

			22 is a term since 4^22 mod 22 = 16.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A036236, A129492, A129493, A129495, A129496, A129497.

Contains A122781 except for 1 and 4.

[k:k in [2..200]| not IsPrime(k)  and  not IsZero(a)  and (PrimeDivisors(a) eq [2]) and  &+[j[1]*j[2]: j in Factorization(a) ] mod 4 eq 0 where a is 4^k mod k]; // Marius A. Burtea, Dec 04 2019

filter:= proc(n) local k,j;
  if isprime(n) then return false fi;
  k:= 4 &^ n mod n;
  j:= padic:-ordp(k,2);
  k>1 and j::even and k = 2^j
end proc:
select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

Select[ Range@ 161, IntegerQ@ Log[4, PowerMod[4, #, # ]] &]

Corrected and extended by R. J. Mathar, May 16 2008

A129495 Composite numbers k such that 5^k (mod k) is a power of 5 greater than 1.

Original entry on oeis.org

10, 15, 20, 26, 30, 34, 38, 40, 46, 50, 56, 58, 60, 62, 65, 74, 78, 82, 86, 94, 100, 106, 118, 120, 122, 124, 129, 130, 132, 134, 140, 141, 142, 143, 146, 150, 155, 158, 159, 166, 177, 178, 182, 183, 190, 194, 195, 200, 201, 202, 206, 213, 214, 217, 218, 219

Offset: 1

Robert G. Wilson v, Apr 17 2007

			26 is a member of the sequence since 5^26 (mod 26) == 25.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036236, A129492, A129493, A129494, A129496, A129497.

Select[Range@ 225, (p = PowerMod[5, #, #]) > 1 && IntegerQ@ Log[5, p] && CompositeQ[#] &] (* corrected by Amiram Eldar, Jul 24 2021 *)

A129496 Composite numbers k such that 6^k (mod k) is a power of 6 greater than 1.

Original entry on oeis.org

10, 15, 21, 30, 35, 38, 42, 45, 46, 58, 60, 62, 70, 74, 82, 84, 86, 90, 94, 105, 106, 118, 122, 126, 132, 134, 140, 142, 146, 158, 166, 178, 180, 182, 185, 190, 194, 202, 206, 210, 214, 215, 217, 218, 219, 222, 226, 228, 231, 237, 249, 252, 254, 258, 259, 262

Offset: 1

Robert G. Wilson v, Apr 17 2007

			38 is a member of the sequence since 6^38 (mod 38) == 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036236, A129492, A129493, A129494, A129495, A129497.

Select[Range@ 266, (p = PowerMod[6, #, #]) > 1 && IntegerQ@ Log[6, p] && CompositeQ[#] &] (* corrected by Amiram Eldar, Jul 24 2021 *)

cp's OEIS Frontend

A178194 Smallest k such that 33^k mod k = n.

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A178202 Smallest k such that 41^k mod k = 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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A128125 Numbers k such that 2^k == 14 (mod k).

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A128370 a(n) = least k such that the remainder of 30^k divided by k is n.

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A129492 Composite numbers k such that 2^k mod k is a power of 2.

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A129493 Composite numbers k such that 3^k mod k is a power of 3.

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Magma

Maple

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A129494 Composite numbers k such that 4^k mod k is a power of 4 greater than 1.

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Magma