A127818 -id:A127818 - cp's OEIS frontend

A036236 Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.

1, 0, 3, 4700063497, 6, 19147, 10669, 25, 9, 2228071, 18, 262279, 3763, 95, 1010, 481, 20, 45, 35, 2873, 2951, 3175999, 42, 555, 50, 95921, 27, 174934013, 36, 777, 49, 140039, 56, 2463240427, 110, 477, 697, 91, 578, 623, 156, 2453, 540923, 55, 70, 345119, 287

Offset: 0

David W. Wilson

a(1) = 0, that is, no n exists with 2^n mod n = 1. Proof: Assume that there exists such n > 1. Consider its smallest prime divisor p. Then 2^n == 1 (mod p) implying that the multiplicative order ord_p(2) divides n. However, since ord_p(2) < p and p is the smallest divisor of n, we have ord_p(2) = 1, that is, p divides 2^1 - 1 = 1 which is impossible. - Max Alekseyev
_Labos Elemer_ asked on Sep 27 2001 if all numbers > 1 eventually appear in A015910, that is, if a(n) > 0 for n > 1.
Ron Graham's conjecture from 1960 states that for any n > 1 there are infinitely many solutions to 2^k mod k = n. - Max Alekseyev, Oct 19 2024
Obviously k > n. - Daniel Forgues, Jul 06 2015

			n = 0: 2^1 mod 1 = 0, a(0) = 1;
n = 1: 2^k mod k = 1, no such k exists, so a(1) = 0;
n = 2: 2^3 mod 3 = 2, a(2) = 3;
n = 3: 2^4700063497 mod 4700063497 = 3, a(3) = 4700063497.

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique, 28, 1980.
R. K. Guy, Unsolved Problems in Number Theory, Section F10.

David W. Wilson, Table of n, a(n) for n = 0..1026 (from the Havermann file)
Joe K. Crump, 2^n mod n
Hans Havermann, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) is unknown
Nick Hobson, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) is unknown. [With 101 new terms and 2 corrections, this supersedes the earlier table from Hans Havermann. The corrections are to terms a(799) and a(881), where smaller values of k were found.]
Nick Hobson, C program
Eric Weisstein's World of Mathematics, 2

Cf. A015910, A015948, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

Bisections: A122182, A124977.

a = Table[0, {75} ]; Do[ b = PowerMod[2, n, n]; If[b < 76 && a[[b]] == 0, a[[b]] = n], {n, 1, 5*10^9} ]; a
(* Second program: *)
t = Table[0, {1000} ]; k = 1; While[ k < 6500000000, b = PowerMod[2, k, k]; If[b < 1001 && t[[b]] == 0, t[[b]] = k]; k++ ]; t
nk[n_] := Module[ {k}, k = 1; While[PowerMod[2, k, k] != n, k++]; k]
Join[{1, 0}, Table[nk[i], {i, 2, 46}]]  (* Robert Price, Oct 11 2018 *)

a(n)=if(n==1,return(0));my(k=n);while(lift(Mod(2,k)^k)!=n,k++);k \\ Charles R Greathouse IV, Oct 12 2011

It's obvious that for each k, a(k) > k and we can easily prove that 2^(3^n) = 3^n-1 (mod 3^n). So 3^n is the least k with 2^k mod k = 3^n-1. Hence for each n, a(3^n-1) = 3^n. - Farideh Firoozbakht, Nov 14 2006

a(3) was first computed by the Lehmers.
More terms from Joe K. Crump (joecr(AT)carolina.rr.com), Sep 04 2000
a(69) = 887817490061261 = 29 * 37 * 12967 * 63809371. - Hagen von Eitzen, Jul 26 2009
Edited by Max Alekseyev, Jul 29 2011

A127821 a(n) = least k such that the remainder of 13^k divided by k is n.

Original entry on oeis.org

2, 11, 5, 51, 44, 7, 15, 371285, 10, 74853, 158, 13757837, 17, 5805311, 22, 2181, 38, 25, 30, 9667, 74, 87, 146, 23441, 88, 19629779, 35, 45, 70, 235433, 46, 55, 34, 309, 134

Offset: 1

Alexander Adamchuk, Jan 30 2007

a(36) > 10^16. - Max Alekseyev, Oct 25 2016

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, Oct 31 2016 [With 207 new terms, this supersedes the earlier table from Robert G. Wilson v et al.]
Doug Strain, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, [With more terms, this supersedes the earlier table from Fausto A. C. Cariboni and Robert G. Wilson v et al.]
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, Feb 06 2007; Nov 30 2010.

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820.

t = Table[0, {10000} ]; k = 1; While[ k < 3500000000, a = PowerMod[13, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

More terms from Robert G. Wilson v, Feb 06 2007

a(264), a(798), a(884), a(896), a(976), a(980), a(152), a(171), a(296), a(464), a(824), a(870) from Daniel Morel, Jun 17, Nov 30 2010

A078457 a(n) = least positive k such that the remainder when 3^k is divided by k is n.

Original entry on oeis.org

1, 2, 2929, 5, 41459, 76, 21, 295, 2352527, 10, 963400369, 1162, 15, 68, 22082967607, 42, 144937, 217, 25, 1054, 1948397, 60, 14495, 266, 721, 28, 4343, 33, 193511, 52, 6884974839, 49, 1055, 48, 622699582951, 39806, 333, 44, 205, 70, 791, 460, 335, 725, 439889

Offset: 0

Robert G. Wilson v, Dec 31 2002

nonn

a(n) > n. Numbers n such that a(n-1) = n are listed in A015949.

a(n) for which no value is currently known: n = 394, 494, 634, 730, 974, 986, 1000, ...

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000 with -1 for large entries where a(n) has not yet been found, Nov 23 2016 [With 162 new terms, this supersedes the earlier table from Robert G. Wilson v et al.]
Jan-Christoph Schlage-Puchta, C program
Robert G. Wilson v et al., Table of n, a(n) for n = 0..10000 with -1 for those entries where a(n) has not yet been found

Cf. A015949, A036236, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

a = Table[0, {50}]; Do[b = PowerMod[3, n, n]; If[b < 51 && a[[b]] == 0, a[[b]] = n], {n, 1, 56*10^6}]; a
t = Table[0, {1000} ]; k = 1; While[ k < 200000000, a = PowerMod[3, k, k]; If[a < 1001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

More terms from Don Reble, Jan 02 2003
a(14) conjectured by Max Alekseyev, Jun 17 2006
a(14) confirmed by Ryan Propper, Feb 03 2007
a(30) from Ryan Propper, Feb 03 2007
a(56), a(110), a(128), a(134), a(187), a(286), a(348), a(392), a(470), a(512), a(550), a(596), a(672), a(676), a(688), a(703), a(716), a(748), a(772), a(784), a(860), a(980) from Jan-Christoph Schlage-Puchta (jcp(AT)mathematik.uni-freiburg.de), May 26 2008
Corrections from Jon E. Schoenfield, Oct 10 2008
a(664), a(928) from Mark Forbes (m.g.forbes(AT)ieee.org), Oct 25 2009
a(34), a(74), a(160) from Hagen von Eitzen, May 08, Jun 16 2009
a(254), a(310) from Daniel Morel, Sep 13, Sep 29 2009
Edited by Max Alekseyev, Feb 11 2012

A119678 a(n) is the least k such that 4^k mod k = n.

Original entry on oeis.org

3, 14, 137243, 5, 6821, 10, 57, 124, 35, 18, 2791496231, 244, 51, 505, 199534799, 20, 30271293169, 49, 45, 236, 399531841, 42, 533, 25, 39, 50, 352957, 36, 995, 98, 33, 112, 47503, 55, 42345881, 44, 2981, 289, 805, 78, 1019971289, 25498, 2121, 212

Offset: 1

Ryan Propper, Jun 12 2006

nonn

a(n) > n.
Numbers n > 1 such that a(n-1) = n are listed in A015950.
a(87) > 10^14.
a(11) <= 2791496231, a(17) <= 140631956671, a(53) <= 52134328061 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 10 2007

Ryan Propper, Table of n, a(n) for n = 1..82
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found

Cf. A015950, A036236, A078457, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

Do[k = 1; While[PowerMod[4, k, k] != n, k++ ]; Print[k], {n, 30}]
t = Table[0, {10000} ]; k = 1; While[ k < 5000000000, a = PowerMod[4, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t (* search limits expanded by Robert G. Wilson v, Jul 14 2009 *)

def a(n):
  k = 1
  while 4**k % k != n: k += 1
  return k
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Mar 14 2021

a(5^k-1) = 5^k.

a(11) = 2791496231 from Robert G. Wilson v, Feb 11 2007; confirmed by Ryan Propper, Feb 15 2007
Link corrected by R. J. Mathar, Jul 24 2009
a(83) = 3085807457009 = 113 * 331 * 82501603 from Hagen von Eitzen, Jul 27 2009

A119679 a(n) = least k such that the remainder when 5^k is divided by k is n.

Original entry on oeis.org

2, 3, 22, 4769, 7, 15853, 114, 9, 28, 35, 14, 1328467, 68, 111, 1555, 9569200211, 76, 2030227, 49, 21, 299, 1097122717, 51, 546707, 26, 27, 121, 529, 596, 3095, 138, 93, 136, 34723, 45, 589, 198, 87, 18142961, 595, 292, 319, 318, 117, 55, 20485243, 91

Offset: 1

Ryan Propper, Jun 12 2006

nonn

From Alexander Adamchuk, Jan 31 2007: (Start)
a(n) > n.
For numbers n such that a(n-1) = n, see A015951 except first term. (End)
a(58) <= 16860204577843069 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 06 2007

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found [This file is now out of date - see the extension lines below. - _N. J. A. Sloane_, May 22 2010]

Cf. A015951, A036236, A078457, A119678, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

Do[k = 1; While[PowerMod[5, k, k] != n, k++ ]; Print[k], {n, 30}]
Table[0, {10000}]; k = 1; lst = {}; While[k < 5000000000, a = PowerMod[5, k, k]; If[ a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a,k}]]; k++ ]; t (* changed (to reflect the new limits) by Robert G. Wilson v, Jul 14 2009 *)

Revised by Max Alekseyev, Sep 25 2007
a(172) = 26598818717 = 23 * 593 * 1039 * 1877, a(288) = 9158745413 = 241 * 347 * 109519, a(518) = 33288260241 = 3 * 43 * 258048529, a(558) = 7722115807 = 7 * 157 * 7026493 from Daniel Morel, May 18 2010
a(848) = 6672480963 = 3 * 241 * 9228881 from Daniel Morel, May 26 2010
a(416) = 10545901269 from Daniel Morel, Jul 05 2010
a(948) = 146246024857 from Daniel Morel, Jul 12 2010
a(822) = 466661006683 from Daniel Morel, Aug 24 2010

A119714 a(n) is the least k such that the remainder when 8^k is divided by k is n.

Original entry on oeis.org

7, 3, 5, 6, 39, 58, 7342733, 9, 36196439, 18, 501, 26, 13607, 249, 119, 20, 33, 25, 866401, 22, 533, 35, 185, 50, 196673, 27, 1843, 36, 69, 34, 551, 55, 3773365, 110, 159, 116, 355, 237, 8401, 52, 471, 81815, 85, 261, 11783479, 3258, 93, 92, 1885511821439

Offset: 1

Ryan Propper, Jun 12 2006

a(61) = 1802190094793 = 11 * 59 * 17839 * 155663. - Hagen von Eitzen, Jul 28 2009

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A127817, A127818, A127819, A127820, A127821.

Do[k = 1; While[PowerMod[8, k, k] != n, k++ ]; Print[k], {n, 48}]
t = Table[0, {10000}]; k = 1; lst = {}; While[k < 4300000000, a = PowerMod[8, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t (* Mathematica coding extended to reflect the new search limits as posted in the a-file by Robert G. Wilson v, Jul 17 2009 *)

a(49) from Hagen von Eitzen, Jul 24 2009

A127817 a(n) = least k such that the remainder when 9^k is divided by k is n.

Original entry on oeis.org

2, 7, 6, 5, 38, 723, 74, 2592842671511, 11, 3827, 14, 717, 34, 59035, 21, 259, 152, 237, 62, 626131, 30, 169, 58, 25, 56, 1921, 39, 361, 65, 49, 63010, 287, 48, 55, 46, 63, 932, 3786791, 69, 69637, 230, 221, 6707, 1057, 57, 4907, 253, 681, 148, 393217991, 70

Offset: 1

Alexander Adamchuk, Jan 30 2007

			For n=4, since 9^5 == 4 (mod 5) and 9^k is not congruent to 4 (mod k) for any k < 5, a(4) = 5. _Michael B. Porter_, Dec 10 2016

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, Nov 21 2016 [With 202 new terms, this supersedes an earlier table of Robert G. Wilson v et al.]

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127818, A127819, A127820, A127821.

a127817 := [seq(0,j=1..nmax)] ; for k from 1 do n := modp(9^k,k) ; if n > 0 and n <= nmax then if op(n,a127817) = 0 then a127817 := subsop(n=k,a127817) ; print( op(1..50,a127817) ) ; fi; fi; od: # R. J. Mathar, Jul 16 2009

t = Table[0, {10000}]; k = 1; lst = {}; While[k < 4500000000, a = PowerMod[9, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t

a(8) <= 2592842671511 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 06 2007
I changed the Mathematica coding to reflect the current limits Robert G. Wilson v, Jul 18 2009
Value for a(8) as suggested by J. K. Crump confirmed by Hagen von Eitzen, Jul 21 2009
Authorship of a-file corrected by R. J. Mathar, Aug 24 2009

A119715 a(n) = least k such that the remainder when 7^k is divided by k is n.

Original entry on oeis.org

2, 5, 46, 339, 22, 387497, 11, 535, 10, 111, 38, 8399, 15, 497, 34, 327, 365, 515, 30, 7219931, 28, 321, 26, 223793, 44, 10718597, 242, 35, 2330, 209, 39, 305, 136, 309, 4382, 10596486211, 45, 24751, 7327, 121, 236, 78821, 55, 4117, 76, 1751, 30514339, 83795, 50, 1333

Offset: 1

Ryan Propper, Jun 12 2006

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found

Cf. A036236, A078457, A119678, A119679, A127816, A119714, A127817, A127818, A127819, A127820, A127821.

t = Table[0, {10000}]; k = 1; lst = {}; While[k < 4100000000, a = PowerMod[7, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t (* changed (to reflect the new limits) by Robert G. Wilson v, Jul 17 2009 *)
lk[n_]:=Module[{k=1},While[PowerMod[7,k,k]!=n,k++];k]; Array[lk,50] (* The program will take a long time to run. *) (* Harvey P. Dale, Jan 29 2023 *)

a(36) = 10596486211 and later terms from Ryan Propper, Feb 02 2007

A127816 a(n) = least k >= 1 such that the remainder when 6^k is divided by k is n.

Original entry on oeis.org

5, 34, 213, 68, 4021227877, 7, 121129, 14, 69, 26, 767, 51, 6191, 22, 201, 20, 1919, 33, 169, 44, 39, 1778, 1926049, 174, 2673413, 50, 63, 451, 1257243481237, 93, 851, 316, 183, 14809, 1969, 38, 1362959, 1826, 177, 289, 65, 87, 5567, 1252, 57, 1651, 6403249

Offset: 1

Alexander Adamchuk, Jan 30 2007, Feb 05 2007

a(7^k-1) = 7^k.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found

Cf. A036236, A078457, A119678, A119679, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

t = Table[0, {10000}]; k = 1; lst = {}; While[k < 5600000000, a = PowerMod[6, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t

a(7^k-1) = 7^k.

a(5) from Joe K. Crump confirmed and a(6)-a(28) added by Ryan Propper, Feb 21 2007
I combined the two Mathematica codings into one and extended the search limits. - Robert G. Wilson v, Jul 16 2009
a(29) as conjectured by J. K. Crump confirmed by Hagen von Eitzen, Jul 21 2009
Corrected authorship of the a-file - R. J. Mathar, Aug 24 2009

A127819 a(n) = least k such that the remainder when 11^k is divided by k is n.

Original entry on oeis.org

2, 3, 118, 7, 39, 155, 38, 9, 14, 18659, 13, 8153, 92, 1317, 106, 35, 26, 49, 34, 57, 68, 253, 327, 1271, 28, 27, 94, 1633, 46, 161, 415, 1141, 44, 909589, 86, 161015, 119, 1293, 82, 673279, 129, 577679, 4578, 77, 164, 65, 74, 3589, 76, 147, 115

Offset: 1

Alexander Adamchuk, Jan 30 2007

nonn

Daniel Morel, Table of n, a(n) for n = 1..189
Daniel Morel, 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. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127820, A127821.

t = Table[0, {10000} ]; k = 1; While[ k < 4300000000, a = PowerMod[11, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t
f[n_]:=Module[{k=1},While[PowerMod[11,k,k]!=n,k++];k]; Array[f,51] (* Harvey P. Dale, Mar 12 2013 *)

a(52) from Max Alekseyev & Joe K. Crump (joecr(AT)carolina.rr.com), Feb 06 2007
a(108), a(514), a(754), a(808), a(820), a(880), a(934), a(948) added by Daniel Morel, May 18 2010, Jun 17 2010
a-file updated by Max Alekseyev, Dec 10 2010
a(112), a(348), a(810) added by Daniel Morel, Mar 12 2015

cp's OEIS Frontend

A036236 Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A127821 a(n) = least k such that the remainder of 13^k divided by k is n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A078457 a(n) = least positive k such that the remainder when 3^k is divided by k is n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A119678 a(n) is the least k such that 4^k mod k = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A119679 a(n) = least k such that the remainder when 5^k is divided by k is n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A119714 a(n) is the least k such that the remainder when 8^k is divided by k is n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A127817 a(n) = least k such that the remainder when 9^k is divided by k is n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A119715 a(n) = least k such that the remainder when 7^k is divided by k is n.

Views