A119678 -id:A119678 - cp's OEIS frontend

A128371 a(n) = least k such that the remainder when 31^k is divided by k is n.

2, 29, 7, 29787, 13, 113413, 51, 23, 11, 3309, 38, 19, 21, 17, 22, 115, 118, 37237, 261, 60212617, 94, 29769, 134, 51205605391, 26, 35, 209, 549, 466, 1558391, 37, 5033228393, 58, 39, 926, 565, 57, 1561, 922, 119, 46, 2512157, 111, 949, 76, 85

Offset: 1

Alexander Adamchuk, Feb 27 2007

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found [Superseded by Cariboni table]
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, Sep 14 2016 [With 123 new terms, this supersedes the earlier table from Robert G. Wilson v et al.]

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A129369, A128370, 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 < 4750000000, a = PowerMod[31, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

More terms from Ryan Propper, Mar 24 2007

a(494) = 14353729267 = 64609 * 222163. a(498) = 9547024387, a(540) = 29711794103. - Daniel Morel, Jun 17 2010. a(618) = 15150617101, a(750) = 13728669221. - Daniel Morel, Jun 28 2010

A128369 a(n) = least k such that the remainder when 29^k is divided by k is n.

Original entry on oeis.org

2, 3, 13, 5, 22, 23, 11, 9, 26, 19, 51, 17, 46, 15, 118, 178523, 152, 92634008921, 102, 24369, 82, 2873, 93, 25, 34, 27, 74, 11227, 31, 39259, 830, 69, 136, 817, 62, 2429, 66, 24351, 802, 121, 184, 3405997613, 714, 45, 398, 5846879, 794, 221, 114

Offset: 1

Alexander Adamchuk, Feb 27 2007

a(408) = 9848641373 = 60343 * 163211.
a(756) = 10012502599 = 11 * 19 * 47906711.
a(886) = 12256265747, a(966) = 10085567837. - Daniel Morel, Jun 08 2010
a(378) = 31113438371, a(492) = 18377996647, a(730) = 22778710711. - Daniel Morel, Jul 05 2010
a(802) = 20290196677, a(826) = 21466370573. - Daniel Morel, Aug 24 2010

Robert G. Wilson v, 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, A129370, 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 < 4000000000, a = PowerMod[29, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

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

Original entry on oeis.org

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

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

A126762 a(n) is the least k > n such that the remainder when n^k is divided by k is n.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 9, 11, 11, 13, 13, 17, 15, 17, 17, 19, 19, 23, 21, 23, 23, 29, 25, 28, 27, 29, 29, 31, 31, 37, 33, 37, 35, 37, 37, 41, 39, 41, 41, 43, 43, 47, 45, 47, 47, 53, 49, 53, 51, 53, 53, 59, 55, 59, 57, 59, 59, 61, 61, 67, 63, 67, 65, 67, 67, 71, 69, 71, 71, 73, 73

Offset: 1

Alexander Adamchuk, Feb 17 2007

nonn

a(n-1) = n for n = {2,3,5,7,9,11,13,15,17,19,21,23,25,27,29,...} = 2 together with odd numbers n > 1.
a(n) coincides with A082048(n) up to n = 24.
a(n) is the smallest number k > n such that n^k == n (mod k). Conjecture: a(n) is the smallest number k > n such that n^(k-1) == 1 (mod k). Thus a(n) is coprime to n. - Thomas Ordowski, Aug 03 2018

Harvey P. Dale, Table of n, a(n) for n = 1..996

Cf. A128149 = Least k such that n^k (mod k) = n-1. Cf. A128172 = Least k such that n^k (mod k) = n+1. Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160. Cf. A082048 = least number greater than n having greater smallest prime factor than that of n.

Table[Min[Select[Range[101],PowerMod[n,#,# ]==n&]],{n,1,100}]
lkgn[n_]:=Module[{k=1},While[PowerMod[n,k,k]!=n,k++];k]; Array[lkgn,80] (* Harvey P. Dale, May 25 2021 *)

Name clarified by Thomas Ordowski, Aug 03 2018

A177495 a(n) is the least k such that the remainder when 100^k is divided by k is n.

Original entry on oeis.org

3, 7, 97, 6, 19, 38, 31, 23, 13, 15, 89, 22, 29, 43, 17, 24, 83, 41, 19003, 580, 79, 42, 1903, 58, 35, 37, 73, 36, 71, 49, 999969, 56, 67, 66, 145, 76, 411, 578, 61, 60, 59, 494, 51, 262, 55, 158, 53, 52, 57, 398, 15673, 69, 1589, 9946, 65, 88, 20940211, 366, 391

Offset: 1

Alexander Adamchuk, May 10 2010

nonn

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160, A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A128369, A128370, A128371, A128372.

Cf. A128149, A128150, A128172.

t = Table[0, {70}]; k = 1; While[k < 210000000, a = PowerMod[100, k, k]; If[a < 71 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t
Table[Module[{k=1},While[PowerMod[100,k,k]!=n,k++];k],{n,60}] (* Harvey P. Dale, Jun 06 2018 *)

A178195 Smallest k such that 34^k mod k = n.

Original entry on oeis.org

1, 3, 1154, 31, 5, 29, 7, 39297, 13, 19055, 18, 23, 22, 21, 535, 19, 20, 62537, 1138, 45, 142, 2092793, 42, 19547, 25, 39279, 50, 749, 36, 39055, 1126, 39, 188, 93641, 35, 634815079, 70, 171, 86, 355, 52, 65387, 713, 69, 148, 253, 74

Offset: 0

Artur Jasinski, May 23 2010

nonn

First unknown term is a(47), where there are no solutions < 2904290724.

For a catalog of sequences of the kind "smallest k such that m^k mod k = n," see A178194.

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

aa = {}; Do[k = 1; While[PowerMod[34, k, k] != n, k++ ]; Print[k]; AppendTo[aa, k], {n, 0, 50}]; aa

A178196 Smallest k such that 35^k mod k = n.

Original entry on oeis.org

1, 2, 3, 26, 31, 10, 29, 14, 9, 13, 55, 57, 23, 92, 21, 22, 19, 99, 187, 134, 2105, 28, 169, 1202, 593791, 30, 27, 1198, 203, 46, 695, 66, 42843, 248, 4023706859, 37, 449467, 132, 327, 1186, 565, 74, 581, 394, 14277, 110, 59867, 62, 1311139, 56, 75

Offset: 0

Artur Jasinski, May 23 2010

nonn

Cf. A036236, A078457, A119678, A119679, A119714, A119715, A119716, A127817-A127821, A128154-A128372, A178194-A178202.

aa = {}; Do[k = 1; While[PowerMod[35, k, k] != n, k++ ]; Print[k]; AppendTo[aa, k], {n, 0, 50}]; aa

Terms a(34) onward from Max Alekseyev, Feb 04 2012

A178197 Smallest k such that 36^k mod k = n.

Original entry on oeis.org

1, 5, 17, 11, 34, 31, 10, 29, 14, 213, 13, 1585, 39, 23, 1282, 21, 20, 19, 142, 56413361, 22, 445, 26, 169, 87, 341, 50, 33, 332, 33607, 57, 55329163, 158, 46623, 1262, 33763, 37, 167987937385549, 74, 123, 284, 12091, 51, 119, 626, 531, 2630, 960641, 104, 473, 98, 75, 116, 424381, 174, 7751, 62, 951, 781, 364789, 206, 545, 1234, 93, 77, 205591, 78, 51367, 614, 159, 1226, 623, 207, 23147, 94, 11847, 100, 3551, 161, 332089, 176, 99, 143, 361841, 202, 73969, 590, 129, 302

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[36, k, k] != n, k++ ]; Print[k]; AppendTo[aa, k], {n, 0, 50}]; aa

Terms a(37) onward from Max Alekseyev, May 07 2012

cp's OEIS Frontend

A128371 a(n) = least k such that the remainder when 31^k is divided by k is n.

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A128369 a(n) = least k such that the remainder when 29^k is divided by k is n.

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

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A126762 a(n) is the least k > n such that the remainder when n^k is divided by k is n.

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A177495 a(n) is the least k such that the remainder when 100^k is divided by k is n.

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A178195 Smallest k such that 34^k mod k = n.

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A178196 Smallest k such that 35^k mod k = n.

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Extensions

A178197 Smallest k such that 36^k mod k = n.

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