A128154 -id:A128154 - cp's OEIS frontend

A128149 Least k such that n^k mod k = n-1.

2929, 137243, 4769, 4021227877, 387497, 7342733, 2592842671511, 22963573117, 18659, 120593747, 13757837, 17651, 17149, 16584420001, 613024059983, 407, 39959, 559, 581831, 305197, 235, 459207143, 855782591, 106709, 17678421233, 240055, 11227

Offset: 3

Alexander Adamchuk, Feb 16 2007

			a(3) = A078457(2) = 2929.

Max Alekseyev, Table of n, a(n) for n = 3..41
Robert G. Wilson v et al., Table of n, a(n) for n = 3..1000 with -1 for large entries where a(n) has not yet been found.

Cf. A128150 = least k such that n^k mod k = (n-1)^2
Cf. A128172 = least k such that n^k mod k = n+1.
Cf. A128148, A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.
Cf. A128154, A128155, A128156, A128157, A128158, A128159, A128160, A128361, A128362, A128363, A128364, A128365, A128366, A128367, A128368, A128369, A129370, A128371, A128372.

t = Table[0, {10000}]; f[n_] := Block[{k = 1}, While[k < 2^23 && PowerMod[n, k, k] + 1 != n, If[ Mod[k, 6] == 1, k += 4, k += 2]]; k]; Do[ If[ t[[n]] == 0, a = f@n; If[a < 2^23, t[[n]] = a; Print[{n, a}]]], {n, 10000}] (* Robert G. Wilson v, Aug 15 2009 *)

a(6) = A127816(5) = 4021227877 found by Ryan Propper, Feb 21 2007
More terms from Alexander Adamchuk, Feb 28 2007
a(9), a(10) from Hagen von Eitzen, Jul 31 2009
More terms from Robert G. Wilson v, Aug 15 2009
a(30), a(35), a(39), a(45) from Max Alekseyev, May 12 2012

A128155 a(n) = least k such that the remainder when 15^k is divided by k is n.

Original entry on oeis.org

2, 13, 6, 11, 10, 533, 218, 119, 12, 145, 214, 57, 106, 17149, 17, 3736136819, 26, 117, 206, 143, 34, 427, 202, 871, 40, 25397, 54, 6877, 52, 115, 194, 6839309, 48, 4857103, 38, 63, 94, 94043, 62, 95, 46, 303, 182, 121214771, 55, 1137417899, 178, 3327, 116

Offset: 1

Alexander Adamchuk, Feb 16 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

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128156, A128157, A128158, A128159, A128160, A128149, A128150.

t = Table[0, {10000} ]; k = 1; While[ k < 3000000000, a = PowerMod[15, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

More terms from Ryan Propper, Feb 28 2007

A128156 a(n) = least k such that the remainder when 16^k is divided by k is n.

Original entry on oeis.org

3, 7, 13, 6, 11, 10, 87, 62, 209, 18, 35, 122, 4083, 22, 16584420001, 17, 1343851, 34, 453, 44, 215, 26, 469, 58, 69, 46, 121, 36, 266461, 49, 813, 56, 19499, 74, 58501, 230, 123, 218, 2077, 78, 17845, 214, 579, 106, 24313642489, 90, 6541, 88, 57, 206

Offset: 1

Alexander Adamchuk, Feb 16 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

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128157, A128158, A128159, A128160, A128149, A128150.

t = Table[0, {10000} ]; k = 1; While[ k < 4100000000, a = PowerMod[16, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

More terms from Ryan Propper, Feb 27 2007

A128157 a(n) = least k such that the remainder when 17^k is divided by k is n.

Original entry on oeis.org

2, 3, 7, 13, 142, 11, 25, 9, 10, 299, 57, 203, 46, 69, 274, 613024059983, 19, 7099195, 30, 21, 134, 24065, 38, 133, 28, 27, 205, 155591, 33, 20452755522967, 49, 165, 35, 391, 99, 94271801, 198, 39, 70, 23353, 62, 2759, 55, 1623, 122, 22649, 665, 1591398755

Offset: 1

Alexander Adamchuk, Feb 16 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

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128158, A128159, A128160, A128149, A128150.

t = Table[0, {10000} ]; k = 1; While[ k < 4500000000, a = PowerMod[17, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

More terms from Hagen von Eitzen, Jul 31 2009
a(338) = 7615772967 = 3 * 11 * 230780999 [From Daniel Morel, May 18 2010]
a(100) = 36706228199, a(154) = 10618746241, a(444) = 10700153359, a(616) = 7969009427, a(720) = 11004291191, a(984) = 11601377453 [From Daniel Morel, Jun 15 2010]
a(184) = 16808380397, a(508) = 34412778035 [From Daniel Morel, Nov 05 2010]

A128158 a(n) = least k such that the remainder when 18^k is divided by k is n.

Original entry on oeis.org

17, 14, 5, 7, 13, 106, 11, 158, 927, 314, 6767, 15, 724317787, 62, 21, 20, 407, 19, 319, 38, 39, 302, 150698261, 30, 1055599, 298, 129, 74

Offset: 1

Alexander Adamchuk, Feb 16 2007

10^15 < a(29) <= 3612834616189533302730621726282897865691021. - Max Alekseyev, Apr 14 2012

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. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128159, A128160, A128149, A128150.

t = Table[0, {10000}]; k = 1; While[k < 3000000000, a = PowerMod[18, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Jun 23 2009 *)

a(13)-a(28) from Robert G. Wilson v, Jun 23 2009

A128159 a(n) = least k such that the remainder when 19^k is divided by k is n.

Original entry on oeis.org

2, 17, 358, 5, 7, 13, 118, 11, 22, 207, 14, 6683, 21, 1055, 221, 6843, 86, 39959, 23, 559, 34, 129, 26, 25, 51, 799, 334, 33, 166, 47427581, 1537, 901, 68, 39, 326, 87169, 44, 161, 46, 3509, 341, 529, 106, 1098179, 158, 657, 314, 49621349, 75, 143, 62, 749, 116

Offset: 1

Alexander Adamchuk, Feb 16 2007

a(447) = 7987803178, a(660) = 11147676413, a(923) = 6246715274. - Daniel Morel, Jun 08 2010

a(216) = 21686254249, a(296) = 40778012377, a(386) = 15891209603, a(582) = 46530896443, a(638) = 15297472657, a(736) = 45211411479, a(872) = 106458212591. - Daniel Morel, Oct 15 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. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128160.

Cf. A128149, A128150.

t = Table[0, {10000} ]; k = 1; While[ k < 3100000000, a = PowerMod[19, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 04 2009 *)
clk=Compile[{{n,Integer}},{k=1};While[PowerMod[19,k,k]!=n,k++];k]; Array[ clk,55] (* _Harvey P. Dale, May 10 2014 *)

More terms from Ryan Propper, Mar 24 2007

More terms from Robert G. Wilson v, Aug 04 2009

A128160 a(n) = least k such that the remainder when 20^k is divided by k is n.

Original entry on oeis.org

19, 3, 17, 6, 15, 7, 13, 9, 11, 18, 7989, 92, 973, 33, 611, 24, 2661, 382, 559, 21, 96641237093, 42, 1887, 94, 155, 27, 60403, 36, 7971, 74, 1172954777, 46, 2470227509, 122, 45, 116, 1837, 362, 779, 60, 469, 358, 1275143, 51, 55, 118, 723, 49

Offset: 1

Alexander Adamchuk, Feb 16 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

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159.

Cf. A128149, A128150.

t = Table[0, {10000} ]; k = 1; While[ k < 4000000000, a = PowerMod[20, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 04 2009 *)

More terms copied from a-file by Hagen von Eitzen, Oct 22 2009

A128372 a(n) = least k such that the remainder when 32^k is divided by k is n.

Original entry on oeis.org

31, 3, 29, 6, 201, 13, 25, 9, 23, 11, 183, 22, 19, 159, 17, 20, 45, 49, 169, 502, 209, 42, 35, 50, 91919, 27, 3265, 36, 1159, 98, 75197, 33, 95, 66, 2817, 38, 1385, 58, 25187, 82, 32727, 982, 55, 117, 7031, 91, 2517, 52, 46528545441593, 57, 503981, 92, 135, 194

Offset: 1

Alexander Adamchuk, Feb 27 2007

Values a(50), ..., a(149) are relatively small again (starting 57, 503981, 92, 135, 194, 576353, 87, 125, 1902, 6019, 323, 43335727, 69, ...). - Hagen von Eitzen, Jun 04 2009

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, A129369, A128370, A128371. 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[32, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* Robert G. Wilson v, Aug 06 2009 *)

Incorrect comment removed by Hagen von Eitzen, Jul 19 2009

a(49) found by Hagen von Eitzen, Jul 20 2009

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

Alexander Adamchuk, Feb 16 2007, May 06 2007

			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.

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.

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

Alexander Adamchuk, Feb 17 2007

nonn

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.]

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

Max Alekseyev, Table of n, a(n) for n = 2..43
Fausto A. C. Cariboni, Table of n, a(n) for n = 2..10000 with -1 for large entries where a(n) has not yet been found, Dec 24 2016 [With 1772 new terms, this supersedes the earlier table from Robert G. Wilson v et al.]
Robert G. Wilson v et al., Table of n, a(n) for n = 2..10000 with -1 for large entries where a(n) has not yet been found

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.

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 *)

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

cp's OEIS Frontend

A128149 Least k such that n^k mod k = n-1.

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A128155 a(n) = least k such that the remainder when 15^k is divided by k is n.

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A128156 a(n) = least k such that the remainder when 16^k is divided by k is n.

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A128157 a(n) = least k such that the remainder when 17^k is divided by k is n.

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A128158 a(n) = least k such that the remainder when 18^k is divided by k is n.

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A128159 a(n) = least k such that the remainder when 19^k is divided by k is n.

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A128160 a(n) = least k such that the remainder when 20^k is divided by k is n.

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A128372 a(n) = least k such that the remainder when 32^k is divided by k is n.

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

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