A015963 -id:A015963 - 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 *)

A116622 Positive integers n such that 13^n == 2 (mod n).

Original entry on oeis.org

1, 11, 140711, 863101, 1856455, 115602923, 566411084209, 706836043419179

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^16. - Max Alekseyev, Nov 02 2018

Cf. A116609.
Solutions to b^n == 2 (mod n): A015919 (b=2), A276671 (b=3), A130421 (b=4), A124246 (b=5), A277401 (b=7), this sequence (b=13), A333269 (b=17).
Solutions to 13^n == k (mod n): A015963 (k=-1), A116621 (k=1), this sequence (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), A116639 (k=15).

Select[Range[1, 500000], Mod[13^#, #] == 2 &] (* G. C. Greubel, Nov 19 2017 *)
Join[{1}, Select[Range[5000000], PowerMod[13, #, #] == 2 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 2; \\ Michel Marcus, Nov 19 2017

One more term from Ryan Propper, Jun 11 2006

Term a(1)=1 is prepended and a(7)-a(8) are added by Max Alekseyev, Jun 29 2011

A116629 Positive integers k such that 13^k == 3 (mod k).

Original entry on oeis.org

1, 2, 5, 166, 287603, 9241538, 2366680105, 8347156585, 21682897793, 6988245760865, 9045859950329, 10076294257985, 50299408064905, 254874726648713

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 24 2017

Some larger terms: 1440926367749746685, 76025040962646716305439353859479569558065. - Max Alekseyev, Jun 29 2011

Cf. A116609, A050259, A122780, A130422, A123061, A277554.

Solutions to 13^n == k (mod n): A001022 (k=0), A015963 (k=-1), A116621 (k=1), A116622 (k=2), this sequence (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), A116639 (k=15).

Join[{1, 2}, Select[Range[1, 5000], Mod[13^#, #] == 3 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1, 2}, Select[Range[10000000], PowerMod[13, #, #] == 3 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 3; \\ Michel Marcus, Nov 19 2017

Two more terms from Ryan Propper, Jan 09 2008

Terms 1,2 are prepended and a(9)-a(14) are added by Max Alekseyev, Jun 29 2011; Nov 24 2017

A116611 Positive integers n such that 13^n == 5 (mod n).

Original entry on oeis.org

1, 2, 4, 44, 82, 236, 25433, 177764, 219244, 86150213, 107218402, 1260236441, 12856300141, 447650116364, 657175627369, 14543842704596, 125035120614917

Offset: 1

Zak Seidov, Feb 19 2006

nonn

No other terms below 10^15.

Some larger terms: 99790373907467602, 846248577183963835642742, 273781047810302314432122404459324, 4174626353309446327489382394518975030641698849116, 211*(13^211-5)/12607932861823674049268705845744 (207 digits). - Max Alekseyev, Jun 29 2011

			44 is in this sequence because 13^44 = 10315908977942302627204470186314316211062255002161 = 234452476771415968800101595143507186615051250049*44 + 5 == 5 (mod 44).

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), this sequence (k=5), A116631 (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609, A128121, A276740, A123091.

Join[{1, 2}, Select[Range[1000000], PowerMod[13, #, #] == 5 &]] (* Robert Price, Apr 10 2020 *)

is(n) = Mod(13,n)^n==5; \\ Charles R Greathouse IV, Jun 08 2015

More terms from Ryan Propper, Apr 01 2006

Terms 1,2,4 are prepended and a(13)-a(17) are added by Max Alekseyev, Jun 29 2011, Nov 27 2017

A116620 Positive integers n such that 13^n == 10 (mod n).

Original entry on oeis.org

1, 3, 9, 74853, 1275039, 27181907, 31261887, 989255061, 4813809711, 3187842157567, 313768710194691

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 06 2018

9909932321420413420533943 is a term. - Max Alekseyev, Jun 29 2011

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), this sequence (k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609.

Join[{1, 3, 9}, Select[Range[2000000], PowerMod[13, #, #] == 10 &]] (* Robert Price, Apr 10 2020 *)

More terms from Ryan Propper, Jun 12 2006

Terms 1,3,9 prepended and a(10)-a(11) added by Max Alekseyev, Jun 29 2011, Nov 06 2018

A116630 Positive integers n such that 13^n == 4 (mod n).

Original entry on oeis.org

1, 3, 51, 129, 125869, 158287, 1723647, 1839003, 90808797, 3661886147, 7368982721, 130424652229, 1616928424359, 4003183891851, 66657658685869

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 26 2017

Some larger terms: 84058689739550643018360088224267, 11083544368708558891212925543084197628431243723. - Max Alekseyev, Jun 26 2011

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

Cf. A116609, A015921, A125949.

Join[{1, 3}, Select[Range[1, 5000], Mod[13^#, #] == 4 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1, 3}, Select[Range[2000000], PowerMod[13, #, #] == 4 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 4; \\ Michel Marcus, Nov 19 2017

More terms from Ryan Propper, Jan 09 2008

Terms 1,3 prepended and a(12)-a(15) added by Max Alekseyev, Jun 26 2011, Nov 26 2017

A116631 Positive integers n such that 13^n == 6 (mod n).

Original entry on oeis.org

1, 7, 8743, 50239, 312389, 8789977, 87453889, 96301009, 3963715129, 5062673539, 6854133309107, 16987071590111, 72278468169733, 411419589731633, 590475819370933

Offset: 1

Zak Seidov, Feb 19 2006

nonn

No other terms below 10^15. A large term: 2455610470454186971078168169. - Max Alekseyev, Dec 19 2017

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), this sequence (k=6), A116632 (k=7), A295532 (k=8), A116636 (k=9), A116620(k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609, A128122, A277628, A277350.

Join[{1}, Select[Range[1, 9000], Mod[13^#, #] == 6 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1}, Select[Range[10000000], PowerMod[13, #, #] == 6 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 6; \\ Michel Marcus, Nov 19 2017

More terms from Ryan Propper, Mar 30 2007

Term 1 is prepended and a(11)-a(15) added by Max Alekseyev, Jun 29 2011, Dec 19 2017

A116636 Positive integers k such that 13^k == 9 (mod k).

Original entry on oeis.org

1, 2, 4, 8, 10, 172, 296, 332, 410, 872, 1048, 1070, 1544, 2830, 3470, 7486, 9196, 22184, 90892, 121174, 299816, 575206, 885112, 1329388, 1386430, 2518994, 4167272, 5600212, 8475016, 9180370, 12348446, 18483076, 19185890, 20806274, 28984094, 37114141

Offset: 1

Zak Seidov, Feb 19 2006

nonn

For k > 9 in this sequence, A116609(k) = 9. - Iain Fox, Nov 20 2017

Max Alekseyev, Table of n, a(n) for n = 1..112

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), this sequence (k=9), A116620 (k=10), A116638 (k=11), A116639 (k=15).

Cf. A116609.

Join[{1, 2, 4, 8}, Select[Range[1, 9000], Mod[13^#, #] == 9 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1,2,4,8},Select[Range[38*10^6],PowerMod[13,#,#]==9&]] (* Harvey P. Dale, Jul 06 2025 *)

isok(n) = Mod(13, n)^n == 9; \\ Michel Marcus, Nov 19 2017

More terms from Ryan Propper, Nov 05 2006

Terms 1,2,4,8 prepended by Max Alekseyev, Jun 28 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.

A116638 Positive integers n such that 13^n == 11 (mod n).

Original entry on oeis.org

1, 2, 158, 301823, 1851103, 8616098, 60528467, 1087582634, 1628818307, 16205558969

Offset: 1

Zak Seidov, Feb 19 2006

No other terms below 10^15. - Max Alekseyev, Nov 07 2018

Large terms: 38763675625170712166, 527929122195463909516681113715859203.

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), this sequence (k=11), A116639 (k=15).

Cf. A116609.

Join[{1, 2}, Select[Range[1, 9000], Mod[13^#, #] == 11 &]] (* G. C. Greubel, Nov 20 2017 *)
Join[{1, 2}, Select[Range[10000000], PowerMod[13, #, #] == 11 &]] (* Robert Price, Apr 10 2020 *)

More terms from Ryan Propper, Jan 11 2008

Edited by Max Alekseyev, May 04 2010

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.

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A116622 Positive integers n such that 13^n == 2 (mod n).

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A116629 Positive integers k such that 13^k == 3 (mod k).

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A116611 Positive integers n such that 13^n == 5 (mod n).

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A116620 Positive integers n such that 13^n == 10 (mod n).

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A116630 Positive integers n such that 13^n == 4 (mod n).

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A116631 Positive integers n such that 13^n == 6 (mod n).

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A116636 Positive integers k such that 13^k == 9 (mod k).

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