A015960 -id:A015960 - 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 *)

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.

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

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

A292336 Numbers k such that k^2 divides 11^k + 1.

Original entry on oeis.org

1, 3, 111, 24753, 5293823142513, 8978700270153, 10982563345209, 28230763238181003

Offset: 1

Max Alekseyev, Sep 15 2017

Cf. A034524. Subsequence of A015960.

Cf. A127092 and A128684.

A333134 Positive integers k such that 11^k == 2 (mod k).

Original entry on oeis.org

1, 3, 413, 1329, 6587, 11629, 75761, 925071199, 9031140861789, 114876097917387, 1314252479257933

Offset: 1

Seiichi Manyama, Mar 20 2020

No other terms below 10^16. Some larger terms: 1584680529929001639, 15598123298097725094806152851164088027801112472240274433891889912569153113. - Max Alekseyev, Jan 07 2025

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=11), A116622 (b=13), A333269 (b=17).

Cf. A015960.

for(k=1, 1e6, if(Mod(11, k)^k==2, print1(k", ")))

a(9)-a(11) from Max Alekseyev, Jan 07 2025

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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A015954 Numbers k such that k | 7^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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A015961 Positive integers k such that k | (12^k + 1).

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A015969 Numbers k that divide 16^k + 1.

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A292336 Numbers k such that k^2 divides 11^k + 1.

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A333134 Positive integers k such that 11^k == 2 (mod k).

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