A128155 -id:A128155

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Search: a128155 -id:a128155

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A128154

a(n) = least k such that the remainder when 14^k is divided by k is n.

+10
34

13, 3, 11, 5, 33, 10, 1967, 9, 23587, 18, 2733, 46, 17651, 15, 93929, 20, 303, 178, 145, 22, 12901, 58, 2721, 25, 17990951, 27, 143, 36, 85, 166, 646123, 82, 2439143677, 55, 63, 76, 319, 123, 295, 52, 51, 77, 247380287953, 45, 5779134947, 90, 87, 74, 175, 146 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Table of n, a(n) for n=1..50.` `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`
	MATHEMATICA	`t = Table[0, {10000} ]; k = 1; While[ k < 3000000000, a = PowerMod[14, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t` `lk[n_]:=Module[{k=1}, While[PowerMod[14, k, k]!=n, k++]; k]; Array[lk, 20] (* Harvey P. Dale, Aug 17 2013 *)`
	CROSSREFS	`Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128155, A128156, A128157, A128158, A128159, A128160, A128149, A128150.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`More terms from Ryan Propper, Feb 28 2007` `a(43) from Hagen von Eitzen, Aug 16 2009`
	STATUS	`approved`

A128149

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

+10
28

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`3,1`
	LINKS	`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.`
	EXAMPLE	`a(3) = A078457(2) = 2929.`
	MATHEMATICA	`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 *)`
	CROSSREFS	`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.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`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`
	STATUS	`approved`

A128156

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

+10
26

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Table of n, a(n) for n=1..50.` `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`
	MATHEMATICA	`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`
	CROSSREFS	`Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128157, A128158, A128159, A128160, A128149, A128150.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`More terms from Ryan Propper, Feb 27 2007`
	STATUS	`approved`

A128157

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

+10
26

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Table of n, a(n) for n=1..48.` `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`
	MATHEMATICA	`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`
	CROSSREFS	`Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128158, A128159, A128160, A128149, A128150.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`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]`
	STATUS	`approved`

A128158

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

+10
26

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`10^15 < a(29) <= 3612834616189533302730621726282897865691021. - Max Alekseyev, Apr 14 2012`
	LINKS	`Table of n, a(n) for n=1..28.` `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`
	MATHEMATICA	`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 *)`
	CROSSREFS	`Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128159, A128160, A128149, A128150.`
	KEYWORD	`hard,more,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`a(13)-a(28) from Robert G. Wilson v, Jun 23 2009`
	STATUS	`approved`

A128159

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

+10
26

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`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`
	LINKS	`Table of n, a(n) for n=1..53.` `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`
	MATHEMATICA	`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 *)`
	CROSSREFS	`Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128160.` `Cf. A128149, A128150.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`More terms from Ryan Propper, Mar 24 2007` `More terms from Robert G. Wilson v, Aug 04 2009`
	STATUS	`approved`

A128160

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

+10
26

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Table of n, a(n) for n=1..48.` `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`
	MATHEMATICA	`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 *)`
	CROSSREFS	`Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128154, A128155, A128156, A128157, A128158, A128159.` `Cf. A128149, A128150.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007`
	EXTENSIONS	`More terms copied from a-file by Hagen von Eitzen, Oct 22 2009`
	STATUS	`approved`

A128372

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

+10
25

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`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`
	LINKS	`Table of n, a(n) for n=1..54.` `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`
	MATHEMATICA	`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 *)`
	CROSSREFS	`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.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 27 2007`
	EXTENSIONS	`Incorrect comment removed by Hagen von Eitzen, Jul 19 2009` `a(49) found by Hagen von Eitzen, Jul 20 2009`
	STATUS	`approved`

A128150

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

+10
23

0, 41459, 35, 9569200211, 2673413, 10596486211, 1885511821439, 235, 12722173, 1971782729, 133617287, 14873, 1465, 1606870609, 4247, 129015968122421, 526673, 835, 1079115301, 12148589879, 12351683, 36947690849, 6385, 5809 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`2,2`
	LINKS	`Table of n, a(n) for n=2..25.`
	EXAMPLE	`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.`
	KEYWORD	`hard,nonn`
	AUTHOR	`Alexander Adamchuk, Feb 16 2007, May 06 2007`
	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`
	STATUS	`approved`

A128172

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

+10
19

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`2,1`
	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.]`
	LINKS	`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`
	EXAMPLE	`a(2) = A036236(3) = 4700063497.`
	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 *)`
	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.`
	KEYWORD	`nonn`
	AUTHOR	`Alexander Adamchuk, Feb 17 2007`
	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`
	STATUS	`approved`

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Last modified August 30 13:06 EDT 2024. Contains 375543 sequences. (Running on oeis4.)