cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A128150 Least k such that n^k mod k = (n-1)^2, or 0 if no such k exists.

Original entry on oeis.org

0, 41459, 35, 9569200211, 2673413, 10596486211, 1885511821439, 235, 12722173, 1971782729, 133617287, 14873, 1465, 1606870609, 4247, 129015968122421, 526673, 835, 1079115301, 12148589879, 12351683, 36947690849, 6385, 5809
Offset: 2

Views

Author

Alexander Adamchuk, Feb 16 2007, May 06 2007

Keywords

Examples

			a(2) = A036236(1) = 0,
a(3) = A078457(2^2) = 41459,
a(4) = A119678(3^2) = 35,
a(5) = A119679(4^2) = 9569200211,
a(6) = A127816(5^2) = 2673413,
a(7) = A119715(6^2) = 10596486211,
a(8) = A119714(7^2) = 1885511821439,
a(9) = A127817(8^2) = 235,
a(10) = A127818(9^2) = 12722173,
a(11) = A127819(100) = 1971782729,
a(12) = A127820(121) = 133617287,
a(13) = A127821(144) = 14873,
a(14) = A128154(169) = 1465,
a(15) = A128155(196) = 1606870609,
a(16) = A128156(225) = 4247,
a(17) = A128157(256) = 129015968122421,
a(18) = A128158(289) = 526673,
a(19) = A128159(324) = 835,
a(20) = A128160(361) = 1079115301,
a(21) = A128361(400) = 12148589879,
a(22) = A128362(441) = 12351683,
a(23) = A128363(484) = 36947690849,
a(24) = A128364(529) = 6385,
a(25) = A128365(576) = 5809,
a(26) = A128366(625) > 10^15,
a(27) = A128367(676) = 299651,
a(28) = A128368(729) > 10^14,
a(29) = A128369(784) = 2645,
a(30) = A128370(841) = 13633321649263,
a(31) = A128371(900) = 1051624907,
a(32) = A128372(961) = 725521, etc.

Crossrefs

Cf. A128148, A128149, A128172, 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.

Extensions

More terms from Alexander Adamchuk, Dec 24 2007
a(13), a(14), a(16), a(18), a(19), a(24), a(25), a(27), a(29), a(32) from Alexander Adamchuk, Feb 17 2008
Corrected A-number in cross-reference. Copied a(8) to a(16) from other sequences. - R. J. Mathar, Aug 08 2009
Edited by Robert G. Wilson v, Aug 20 2009
a(17) from Joe Crump (joecr(AT)carolina.rr.com), Sep 17 2009.
More terms and general editing from Robert G. Wilson v, Sep 30 2009
a(20)-a(22) from Robert G. Wilson v, Oct 17 2009
a(23), a(30) from Max Alekseyev, Feb 11, Mar 31 2010

A128172 Least k such that n^k mod k = n + 1.

Original entry on oeis.org

4700063497, 41459, 6821, 15853, 121129, 535, 36196439, 3827, 15084115509707, 8153, 20395, 5805311, 93929, 3736136819, 1343851, 7099195, 319, 559, 96641237093, 5053, 1535, 280517, 148731221, 869, 2062919, 17473, 803, 39259
Offset: 2

Views

Author

Alexander Adamchuk, Feb 17 2007

Keywords

Comments

a(n)=k must be odd since n and n+1 are of opposite parity. The only way this can occur is if k is odd. - Robert G. Wilson v, Aug 12 2009 [Comment corrected by Fausto A. C. Cariboni, Nov 20 2016.]

Examples

			a(2) = A036236(3) = 4700063497.

Links

Crossrefs

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159, A128160.
Cf. A128149 = Least k such that n^k mod k = n - 1.

Programs

  • Mathematica
    t = Table[0, {10000}]; f[n_] := Block[{k = 1}, While[k < 2097153 && PowerMod[n, k, k] != n + 1, If[ Mod[k, 6] == 1, k += 4, k += 2]]; k]; Do[ If[ t[[n]] == 0, a = f@n; If[a < 2097153, t[[n]] = a; Print[{n, a}]]], {n, 10000}]; t (* Robert G. Wilson v, Aug 12 2009 *)

Extensions

a(15) = A128155(16) = 3736136819 and a(16) = A128156(17) = 1343851 found by Ryan Propper, Feb 27-28 2007
a(10), a(17), a(20), a(23)-a(24), a(26), a(30)-a(31), a(33)-a(35) determined by Tyler Cadigan (tylercadigan(AT)gmail.com), Feb 21 2009
Terms corrected by Hagen von Eitzen and R. J. Mathar, Aug 05 2009
Obsolete link to a-file duplicate removed by R. J. Mathar, Aug 24 2009
Edited and a(36), a(38), a(41), a(48), a(49) added by Max Alekseyev, Feb 04, Mar 25, May 07 2012

A128361 a(n) = least k such that the remainder when 21^k is divided by k is n.

Original entry on oeis.org

2, 19, 6, 17, 218, 15, 14, 13, 12, 11, 86, 9249, 214, 133, 69, 4084085, 106, 39, 422, 581831, 23, 5053, 38, 9237, 26, 775, 46, 1253, 206, 51, 82, 671, 34, 617741981, 58, 45, 202, 289, 87, 6401, 185, 217, 341, 3485351, 66, 2718013, 394, 111, 56, 8064317, 75
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A128362, A128363, A128364, A128365, A128366, A128367, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000}]; k = 1; While[k < 3000000000, a = PowerMod[21, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k]; k++ ]; t (* Robert G. Wilson v, Jun 25 2009 *)
lk[n_]:=Module[{k=1},While[PowerMod[21,k,k]!=n,k++];k]; Array[lk,60] (* The program takes a long time to run *) (* Harvey P. Dale, Oct 22 2016 *)

Extensions

a(16) - a(51) from Robert G. Wilson v, Jun 25 2009

A128362 a(n) = least k such that the remainder when 22^k is divided by k is n.

Original entry on oeis.org

3, 5, 19, 6, 17, 478, 25, 14, 13, 18, 187, 118, 15, 94, 1032913, 20, 64311245, 466, 3543, 58, 305197, 23, 1535, 46, 10623, 458, 5099785, 36, 2723, 454, 10617, 226, 55, 87, 35459, 140, 45, 446, 1373093, 51, 3604637279, 65, 75, 110, 23299, 57, 305, 52, 10599
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A128361, A128363, A128364, A128365, A128366, A128367, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000}]; k = 1; While[k < 4000000000, a = PowerMod[22, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Jun 26 2009 *)

Extensions

a(15) - a(40) from Robert G. Wilson v, Jun 26 2009
a(41) - a(49) from Robert G. Wilson v, Jun 27 2009

A128363 a(n) = least k such that the remainder when 23^k is divided by k is n.

Original entry on oeis.org

2, 3, 5, 19, 262, 17, 58, 9, 10, 13, 14, 55, 86, 12153, 514, 111823, 95, 25, 30, 12147, 68, 235, 29, 280517, 56, 27, 502, 16805, 51, 49, 166, 35, 62, 1837, 38, 977969, 82, 39, 1370, 289, 122, 9822698929535, 65, 133, 697, 161, 303, 19445, 50, 147, 259, 1247
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A128361, A128362, A128364, A128365, A128366, A128367, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000} ]; k = 1; While[ k < 4000000000, a = PowerMod[23, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 04 2009 *)
Table[Module[{k=1},While[PowerMod[23,k,k]!=n,k++];k],{n,35}] (* The program generates the first 35 terms of the sequence. *) (* Harvey P. Dale, Jul 18 2025 *)

Extensions

a(42), a(64) from Hagen von Eitzen, Aug 04 2009
a(750), a(770), a(234), a(274), a(406), a(600), a(610), a(754) from Daniel Morel, May 31, Aug 24, Sep 20 2010
a(84) from Max Alekseyev, Apr 13 2012

A128364 a(n) = least k such that the remainder when 24^k is divided by k is n.

Original entry on oeis.org

23, 11, 7, 5, 19, 10, 17, 142, 15, 566, 13, 78, 5865637, 205, 13809, 20, 589, 39, 35, 278, 129, 554, 459207143, 25, 148731221, 50, 63, 274, 2855, 33, 49, 34, 5429, 542, 5528521301, 42, 2773, 538, 185, 77, 3220589, 66, 90553, 956, 2317, 70, 161, 104
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A128361, A128362, A128363, A128365, A128366, A128367, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000}]; k = 1; While[k < 3600000000, a = PowerMod[25, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Jun 29 2009 *)
lk[n_] := Module[{k = 1}, While[PowerMod[24, k, k] != n, k++]; k]; Array[ lk,48] (* Harvey P. Dale, Jan 18 2019 *)

Extensions

a(13) - a(34) from Robert G. Wilson v, Jun 29 2009

A128365 a(n) = least k such that the remainder when 25^k is divided by k is n.

Original entry on oeis.org

2, 23, 11, 7, 10, 19, 57, 17, 14, 15, 614, 13, 34
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Comments

a(14) > 10^15. - Max Alekseyev, Apr 14 2012

Links

Crossrefs

Cf. A128361, A128362, A128363, A128364, A128366, A128367, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000} ]; k = 1; While[ k < 4000000000, a = PowerMod[25, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 04 2009 *)

A128366 a(n) = least k such that the remainder when 26^k is divided by k is n.

Original entry on oeis.org

5, 3, 23, 6, 7, 10, 19, 9, 17, 18, 15, 92, 18881, 319, 36091, 20, 203, 94, 49, 21, 42395, 42, 17553, 326, 106709, 27, 2062919, 36, 14099, 34, 35, 46, 850984699, 214, 5847, 44, 341, 58, 377, 106, 105, 634, 301265879, 158, 93107, 90, 759, 176, 187, 69, 685, 78
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Comments

If a(199) exists, a(199) >= 148000000000. - Zhuorui He, Jul 24 2025

Links

Crossrefs

Cf. A128361, A128362, A128363, A128364, A128365, A128367, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000}]; k = 1; lst = {}; While[k < 1200000000, a = PowerMod[26, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]; If[a + 1 == k, AppendTo[lst, a]; Print@lst]]; k++ ]; lst (* Robert G. Wilson v, Jun 30 2009 *)

Extensions

a(27)-a(52) from Robert G. Wilson v, Jun 30 2009

A128367 a(n) = least k such that the remainder when 27^k is divided by k is n.

Original entry on oeis.org

2, 5, 6, 23, 11, 7, 25, 19, 10, 17, 718, 165, 35, 533, 33, 3738251, 178, 57, 142, 9779, 60, 2227273193, 55, 19659, 724, 17678421233, 29, 17473, 70, 19653, 209, 3005, 48, 28777, 694, 111, 346, 1441, 46, 15977, 86, 3399, 12614, 4387, 116, 527
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128368, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000} ]; k = 1; While[ k < 4500000000, a = PowerMod[27, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

Extensions

a(16) - a(25) from Robert G. Wilson v, Aug 06 2009

A128368 a(n) = least k such that the remainder when 28^k is divided by k is n.

Original entry on oeis.org

3, 13, 5, 6, 23, 11, 15, 194, 19, 18, 17, 148, 213, 22, 131209, 20, 2335, 25, 7311, 44, 259, 51, 5263, 38, 21927, 758, 240055, 29, 803, 58, 21921, 55, 4405, 39, 413, 316, 549, 746, 17831, 62, 4367, 106, 165, 74, 19253, 82, 6455, 88, 147, 734, 62093, 122
Offset: 1

Views

Author

Alexander Adamchuk, Feb 27 2007

Keywords

Links

Crossrefs

Cf. A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128369, 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.

Programs

  • Mathematica
    t = Table[0, {10000} ]; k = 1; While[ k < 5000000000, a = PowerMod[28, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)
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