A015950 -id:A015950 - cp's OEIS frontend

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

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

A015949 Numbers k such that k | 3^k + 1.

Original entry on oeis.org

1, 2, 10, 50, 250, 1250, 5050, 6250, 11810, 25250, 31250, 59050, 126250, 156250, 295250, 510050, 631250, 750250, 781250, 1476250, 2125250, 2550250, 3156250, 3751250, 3906250, 5964050, 7381250, 10626250, 12751250, 13947610, 15781250

Offset: 1

Robert G. Wilson v

nonn

a(n) mod 20 = 10 for n >= 3. - G. C. Greubel, Nov 05 2018
This sequence is infinite, because for n > 1, 3^a(n) + 1 is in this sequence. - Jinyuan Wang, Nov 06 2018
For the provided data, if k is a term then p*k is a term where p is an odd divisor of k. - David A. Corneth, Nov 06 2018

Giovanni Resta, Table of n, a(n) for n = 1..180 (first 100 terms from G. C. Greubel)

Cf. A034472 (3^n+1).
Cf. A006521 (k | 2^k + 1), A015950 (k | 4^k + 1), A015951 (k | 5^k + 1).
Column k=3 of A333429.

[n: n in [1..2*10^7]| Modexp(3, n, n)+1 eq n]; // Vincenzo Librandi, Nov 01 2018

Do[If[PowerMod[3, n, n] + 1 == n, Print[n]], {n, 1, 10^7}] (* Jinyuan Wang, Nov 01 2018 *)
Select[Range[16*10^6],PowerMod[3,#,#]==#-1&] (* Harvey P. Dale, Dec 15 2024 *)

for(n=1, 10^7, if(Mod(3, n)^n==-1, print1(n, ", "))) \\ Jinyuan Wang, Nov 01 2018

Corrected by David W. Wilson

A333429 A(n,k) is the n-th number m that divides k^m + 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 1, 2, 9, 0, 1, 5, 10, 27, 0, 1, 2, 25, 50, 81, 0, 1, 7, 3, 125, 250, 171, 0, 1, 2, 49, 9, 205, 1250, 243, 0, 1, 3, 10, 203, 21, 625, 5050, 513, 0, 1, 2, 9, 50, 343, 26, 1025, 6250, 729, 0, 1, 11, 5, 27, 250, 1379, 27, 2525, 11810, 1539, 0

Offset: 1

Alois P. Heinz, Mar 20 2020

			Square array A(n,k) begins:
  1,    1,     1,    1,   1,    1,     1,   1,    1,     1, ...
  2,    3,     2,    5,   2,    7,     2,   3,    2,    11, ...
  0,    9,    10,   25,   3,   49,    10,   9,    5,   121, ...
  0,   27,    50,  125,   9,  203,    50,  27,   25,   253, ...
  0,   81,   250,  205,  21,  343,   250,  57,   82,  1331, ...
  0,  171,  1250,  625,  26, 1379,  1250,  81,  125,  2783, ...
  0,  243,  5050, 1025,  27, 1421,  2810, 171,  625,  5819, ...
  0,  513,  6250, 2525,  63, 2401,  5050, 243, 2525, 11891, ...
  0,  729, 11810, 3125,  81, 5887,  6250, 513, 3125, 14641, ...
  0, 1539, 25250, 5125, 147, 9653, 14050, 729, 3362, 30613, ...

Alois P. Heinz, Antidiagonals n = 1..20, flattened

Columns k=1-16 give: A130779 (for n>=1), A006521, A015949, A015950, A015951, A015953, A015954, A015955, A015957, A015958, A015960, A015961, A015963, A015965, A015968, A015969.
Rows n=1-2 give: A000012, A092067.
Main diagonal gives A333430.
Cf. A333432.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=1 then `if`(n<3, n, 0) else
        while nops(p(k)) 0 do od;
          p(k):= [p(k)[], h]
        od; p(k)[n] fi
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

dmax = 12;
mmax = 2^(dmax+3);
col[k_] := col[k] = Select[Range[mmax], Divisible[k^#+1, #]&];
A[n_, k_] := If[n>2 && k==1, 0, col[k][[n]]];
Table[A[n, d-n+1], {d, 1, dmax}, {n, 1, d}] // Flatten (* Jean-François Alcover, Jan 05 2021 *)

A015954 Numbers k such that k | 7^k + 1.

Original entry on oeis.org

1, 2, 10, 50, 250, 1250, 2810, 5050, 6250, 14050, 25250, 31250, 40210, 70250, 126250, 156250, 201050, 351250, 510050, 631250, 650050, 781250, 789610, 1005250, 1265050, 1419050, 1756250, 2550250, 3156250, 3250250, 3906250, 3948050, 5026250, 6325250, 7095250, 8781250, 9478130

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..100

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), this sequence (b=7), A015955 (b=8), A015957 (b=9), A015958 (b=10), A015960 (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).
Cf. A277370, A277401.
Column k=7 of A333429.

A015960 Numbers k such that k | 11^k + 1.

Original entry on oeis.org

1, 2, 3, 9, 27, 81, 111, 122, 243, 333, 729, 999, 2187, 2997, 4107, 6561, 7442, 8991, 10233, 12321, 13203, 19683, 24753, 26973, 30699, 36963, 39609, 59049, 74259, 80919, 89426, 92097, 110889, 118341, 118827, 151959, 177147, 222777

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..200

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), A015954 (b=7), A015955 (b=8), A015957 (b=9), A015958 (b=10), this sequence (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).
Cf. A333134.
Column k=11 of A333429.

Select[Range[250000],PowerMod[11,#,#]==#-1&] (* Harvey P. Dale, Nov 09 2022 *)

A015953 Numbers k such that k | 6^k + 1.

Original entry on oeis.org

1, 7, 49, 203, 343, 1379, 1421, 2401, 5887, 9653, 9947, 11977, 16807, 39991, 41209, 67571, 69629, 83839, 117649, 170723, 271663, 279937, 288463, 347333, 472997, 487403, 586873, 706643, 823543, 1159739, 1195061, 1901641, 1959559, 2019241, 2359469, 2431331

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..100

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), this sequence (b=6), A015954 (b=7), A015955 (b=8), A015957 (b=9), A015958 (b=10), A015960 (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).

Column k=6 of A333429.

Select[Range[2000000],PowerMod[6,#,#]==#-1&] (* Harvey P. Dale, Aug 28 2012 *)

A015955 Numbers k such that k | 8^k + 1.

Original entry on oeis.org

1, 3, 9, 27, 57, 81, 171, 243, 513, 729, 1083, 1539, 2187, 3249, 4401, 4617, 6561, 9747, 13203, 13851, 19683, 20577, 29241, 32547, 39609, 41553, 59049, 61731, 83619, 87723, 97641, 118179, 118827, 124659, 177147, 185193, 250857, 263169

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..200

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), A015954 (b=7), this sequence (b=8), A015957 (b=9), A015958 (b=10), A015960 (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).

Column k=8 of A333429.

A015957 Numbers k such that k | 9^k + 1.

Original entry on oeis.org

1, 2, 5, 25, 82, 125, 625, 2525, 3125, 3362, 5905, 12625, 15625, 29525, 63125, 78125, 137842, 147625, 188354, 255025, 315625, 375125, 390625, 738125, 1062625, 1275125, 1578125, 1875625, 1953125, 2982025, 3690625, 5313125, 5651522, 6375625, 6973805

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..200

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), A015954 (b=7), A015955 (b=8), this sequence (b=9), A015958 (b=10), A015960 (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).

Column k=9 of A333429.

Select[Range[7*10^6],PowerMod[9,#,#]==#-1&] (* Harvey P. Dale, Apr 21 2024 *)

More terms from David W. Wilson

A015958 Numbers k such that k | 10^k + 1.

Original entry on oeis.org

1, 11, 121, 253, 1331, 2783, 5819, 11891, 14641, 30613, 35167, 45023, 64009, 96569, 130801, 133837, 161051, 273493, 336743, 386837, 495253, 527197, 558877, 640343, 704099, 808841, 1035529, 1062259, 1438811, 1472207, 1652849, 1771561, 2221087, 3008423, 3045449

Offset: 1

Robert G. Wilson v

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..100

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), A015954 (b=7), A015955 (b=8), A015957 (b=9), this sequence (b=10), A015960 (b=11), A015961 (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).

Column k=10 of A333429.

Select[Range[15*10^5],PowerMod[10,#,#]==#-1&] (* Harvey P. Dale, Oct 01 2017 *)

Corrected by T. D. Noe, Oct 31 2006

A015961 Positive integers k such that k | (12^k + 1).

Original entry on oeis.org

1, 13, 169, 1027, 2197, 13351, 28561, 81133, 173563, 371293, 468481, 685633, 1054729, 2256319, 2890927, 4826809, 6090253, 6409507, 8913229, 13711477, 29332147, 37009999, 37582051, 54165007, 62748517, 79173289, 83323591, 115871977, 178249201, 228383233

Offset: 1

Robert G. Wilson v

nonn

Solutions to b^k == -1 (mod k): A006521 (b=2), A015949 (b=3), A015950 (b=4), A015951 (b=5), A015953 (b=6), A015954 (b=7), A015955 (b=8), A015957 (b=9), A015958 (b=10), A015960 (b=11), this sequence (b=12), A015963 (b=13), A015965 (b=14), A015968 (b=15), A015969 (b=16).

Column k=12 of A333429.

More terms from Max Alekseyev, Aug 01 2011

a(30) from Jon E. Schoenfield, Aug 27 2021

cp's OEIS Frontend

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

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A015949 Numbers k such that k | 3^k + 1.

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A333429 A(n,k) is the n-th number m that divides k^m + 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A015954 Numbers k such that k | 7^k + 1.

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A015960 Numbers k such that k | 11^k + 1.

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A015953 Numbers k such that k | 6^k + 1.

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A015955 Numbers k such that k | 8^k + 1.

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A015957 Numbers k such that k | 9^k + 1.

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A015958 Numbers k such that k | 10^k + 1.

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