A128149 -id:A128149 - cp's OEIS frontend

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

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

A128148 a(n) = least k such that 3^k mod k = 2^n.

Original entry on oeis.org

2, 2929, 41459, 2352527, 144937, 1055, 1829903, 7316185805, 114491, 3146746271, 5028467, 20299, 69609309001, 129433, 15307006153, 2149705, 66469, 559182815, 18429503, 4529951, 7094711, 83591212702535, 1251548749, 38088889

Offset: 0

Alexander Adamchuk, Feb 16 2007

			a(1) = A128149(3) = 2929.
a(2) = A128150(3) = 41459.

Cf. A078457 = least k such that the remainder when 3^k is divided by k is n.

Cf. A036236, A128149, A128150.

a(n) = A078457(2^n).

a(7)-a(9) from A078457. Max Alekseyev, Mar 11 2009
Extended by Max Alekseyev, Mar 15 2009
a(20) from Hagen von Eitzen, Aug 01 2009
a(21)-a(23) from Max Alekseyev, Feb 13 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 *)

A321364 Positive integers m such that 13^m == 12 (mod m).

Original entry on oeis.org

1, 13757837, 6969969233, 514208575135

Offset: 1

Max Alekseyev, Nov 07 2018

No other terms below 10^15.

Some larger terms: 14551705803598782884189, 268766423508299769671017810348321281664525668552158231.

Solutions to 13^m == k (mod m): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), this sequence (k=12), A321365 (k=14), A116639 (k=15).

Cf. A116609, A127821, A128149.

is(n) = Mod(13, n)^n==12 \\ Felix Fröhlich, Nov 08 2018

A321365 Positive integers n such that 13^n == 14 (mod n).

Original entry on oeis.org

1, 5805311, 392908759, 399614833907, 2674764845549, 21997277871211, 67146783889057

Offset: 1

Max Alekseyev, Nov 08 2018

No other terms below 10^15.

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), A116629 (k=3), A116630 (k=4), A116611 (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A321364 (k=12), this sequence (k=14), A116639 (k=15).

Cf. A116609, A127821, A128149, A128172.

is(n) = Mod(13, n)^n==14; \\ Felix Fröhlich, Nov 08 2018

A177496 a(n) is the least k such that the remainder when 1000^k is divided by k is n.

Original entry on oeis.org

3, 62, 997, 6, 115, 7, 51, 14, 991, 11, 23, 13, 21, 17, 197, 24, 983, 158, 109, 35, 89, 42, 977, 61, 39, 34, 139, 36, 971, 38, 3291, 188, 967, 66, 193, 92, 57, 74, 999161, 52, 137, 479, 69, 239, 191, 53, 953, 49, 317, 70, 73, 79, 947, 65291, 63, 59, 448991, 114, 941

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, A177495.

Cf. A128149, A128150, A128172.

t = Table[0, {98}]; k = 1; While[k < 10000000, a = PowerMod[1000, k, k]; If[a < 99 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t
lk[n_]:=Module[{k=1},While[PowerMod[1000,k,k]!=n,k++];k]; Array[lk,60] (* Harvey P. Dale, Jul 21 2021 *)

cp's OEIS Frontend

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

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A128148 a(n) = least k such that 3^k mod k = 2^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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A321364 Positive integers m such that 13^m == 12 (mod m).

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A321365 Positive integers n such that 13^n == 14 (mod n).

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A177496 a(n) is the least k such that the remainder when 1000^k is divided by k is n.

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