A015961 -id:A015961 - cp's OEIS frontend

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.

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 *)

A014951 Positive integers k such that k divides 12^k - 1.

Original entry on oeis.org

1, 11, 121, 253, 1331, 2783, 5819, 11891, 14641, 30613, 64009, 130801, 133837, 161051, 273493, 336743, 558877, 704099, 768361, 849299, 1438811, 1472207, 1771561, 3008423, 3078251, 3544453, 3704173, 6147647, 6290339, 7745089

Offset: 1

Olivier Gérard

nonn

Also, the integers k such that k divides s(k), where s(1)=1 and s(j) = s(j-1) + j*12^(j-1).
From Alexander Adamchuk, May 16 2010: (Start)
11 divides a(n) for n > 1.
All powers of 11 are terms.
Prime divisors of a(n) in order of appearance: {11, 23, 47, 3037, 7019, 2663, 5171, 47917, 12973, ...}. (End)

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

Cf. A015961.

Join[{1},Select[Range[775*10^4],PowerMod[12,#,#]==1&]] (* Harvey P. Dale, Jun 08 2022 *)

a(11)-a(30) from Alexander Adamchuk, May 16 2010

Edited by Max Alekseyev, May 20 2011

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

A015969 Numbers k that divide 16^k + 1.

Original entry on oeis.org

1, 17, 289, 4913, 83521, 1419857, 6029713, 12027313, 24137569, 85525793, 102505121, 204464321, 410338673, 1453938481, 1742587057, 3475893457, 6975757441, 24716954177, 29623979969, 59090188769, 111612202577, 118587876497, 420188221009, 500540685121, 503607659473

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..34 (terms < 10^13)

Cf. A006521, A015949, A015950, A015951, A015953, A015954, A015955, A015957, A015958, A015960, A015961, A015963, A015965, A015968, A014957.

Column k=16 of A333429.

More terms from Max Alekseyev, Oct 02 2010

Missing terms a(10), a(14), a(18), and a(23) from Giovanni Resta, Mar 23 2020

A292337 Numbers k such that k^2 divides 12^k + 1.

Original entry on oeis.org

1, 13, 1027, 468481, 2890927, 37009999, 228383233, 2126334691, 104180336299, 8230246567621, 34238014256797, 76626723259567, 131147819778859, 219609509027857, 472852182249889, 1033753383066391, 2704803126286963, 10360677762529861, 16436880348770011, 17349151213200703, 81666517262244889, 686779630199762359

Offset: 1

Max Alekseyev, Sep 16 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..29

Subsequence of A015961.

Cf. A128405, A128685.

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A014951 Positive integers k such that k divides 12^k - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A015969 Numbers k that divide 16^k + 1.

Views

Author

Keywords

Links

Crossrefs

Extensions

A292337 Numbers k such that k^2 divides 12^k + 1.

Views

Author

Keywords

Links

Crossrefs