A333429 -id:A333429 - cp's OEIS frontend

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

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

A015968 Positive integers k such that k | (15^k + 1).

Original entry on oeis.org

1, 2, 226, 25538, 2885794, 326094722, 36848703586, 4163903505218, 470521096089634, 53168883858128642, 638026606623638426, 6008083875968536546, 6789793858544278594

Offset: 1

Robert G. Wilson v

No other terms below 10^19.

Some larger terms: 15878365178317113816706, (15^226+1)/238519527370488646998297200411598927857 (228 digits). - Max Alekseyev, Aug 03 2011

Column k=15 of A333429.

a(6)-a(13) from Max Alekseyev, Oct 02 2010, Aug 03 2011, Sep 17 2017

A333432 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, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 9, 8, 0, 6, 1, 5, 4, 21, 16, 0, 7, 1, 2, 25, 6, 27, 20, 0, 8, 1, 7, 3, 125, 8, 63, 32, 0, 9, 1, 2, 49, 4, 625, 12, 81, 40, 0, 10, 1, 3, 4, 343, 6, 1555, 16, 147, 64, 0, 11, 1, 2, 9, 8, 889, 8, 3125, 18, 171, 80, 0, 12

Offset: 1

Seiichi Manyama, Mar 21 2020

			Square array A(n,k) begins:
  1, 1,  1,   1,  1,     1,  1,     1,  1, ...
  2, 0,  2,   3,  2,     5,  2,     7,  2, ...
  3, 0,  4,   9,  4,    25,  3,    49,  4, ...
  4, 0,  8,  21,  6,   125,  4,   343,  8, ...
  5, 0, 16,  27,  8,   625,  6,   889, 10, ...
  6, 0, 20,  63, 12,  1555,  8,  2359, 16, ...
  7, 0, 32,  81, 16,  3125,  9,  2401, 20, ...
  8, 0, 40, 147, 18,  7775, 12,  6223, 32, ...
  9, 0, 64, 171, 24, 15625, 16, 16513, 40, ...

Seiichi Manyama, Antidiagonals n = 1..25, flattened
OEIS Wiki, 2^n mod n

Columns k=1-20 give: A000027, A063524, A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A333506, A128360.
Main diagonal gives A333433.
Cf. A333429, A333500.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=2 then `if`(n=1, 1, 0) else
        while nops(p(k)) 1 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);  # Alois P. Heinz, Mar 24 2020

A[n_, k_] := Module[{h, p}, p[_] = {1}; If[k == 2, If[n == 1, 1, 0], While[ Length[p[k]] < n, For[h = 1 + p[k][[-1]], Mod[k^h, h] != 1, h++]; p[k] = Append[p[k], h]]; p[k][[n]]]];
Table[A[n, 1+d-n], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

A092067 a(n) is the smallest number m such that m > 1 and m divides n^m + 1.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2, 83, 2, 5, 2, 3, 2, 89, 2, 7, 2, 3, 2, 5, 2, 97

Offset: 1

Farideh Firoozbakht, Mar 28 2004

nonn

a(n)=2 iff n is odd. If n is even then every prime factor of n+1 is a solution of the equation (n^x + 1) mod x = 0, and if n is odd, the smallest prime factor of n+1 (2) is a solution of (n^x + 1) mod x = 0, so for each n, a(n) is not greater than the smallest prime factor of n+1.
Conjecture 1: All terms of this sequence are primes. We know if n is odd a(n) is the smallest prime factor of n+1.
Conjecture 2: For each n, a(n) is the smallest prime factor of n+1 or a(n)=A020639(n+1).
From Charlie Neder, Jun 16 2019: (Start)
Theorem: a(n) = A020639(n+1).
Proof: If a(n) is composite (kp, say) then n^(kp) == -1 (mod p), but then n^k is also congruent to -1 (mod p) by Fermat's little theorem, contradicting the assumption that a(n) was minimal. Thus, a(n) must be prime, and using Fermat's little theorem again shows that n^p == -1 (mod p) iff n == -1 (mod p), and A020639(n+1) gives the least p such that this is the case. (End)
The theorem plus the conjecture 2 in A092028 imply a(n) = A092028(n+2). - R. J. Mathar, Mar 21 2023

			a(6)=7 because 7 divides 6^7 + 1 and there doesn't exist m such that 1 < m < 7 and m divides 6^m + 1.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A020639, A092028.

Row n=2 of A333429.

a[n_] := (For[k=2, Mod[n^k+1, k]>0, k++ ];k); Table[a[n], {n, 100}]
snm[n_]:=Module[{m=2},While[PowerMod[n,m,m]!=m-1,m++];m]; Array[snm,100] (* Harvey P. Dale, Jul 31 2021 *)

A333430 a(n) is the n-th number m that divides n^m + 1 (or 0 if m does not exist).

Original entry on oeis.org

1, 3, 10, 125, 21, 1379, 2810, 243, 3125, 30613, 729, 685633, 850, 183

Offset: 1

Alois P. Heinz, Mar 20 2020

a(15) > 10^19.
a(16) = 29623979969.
a(17) = 250.
a(19) = 13915.
a(15) <= 76717223012242243155874. - Jinyuan Wang, Mar 25 2020
From Jon E. Schoenfield, Aug 28 2021: (Start)
a(18) = 16983563041.
Next 20 terms after the first unknown term (a(15)): 29623979969, 250, 16983563041, 13915, 1143, 23426, 5608987, 2187, 75625, 25160213, 2709, 26803, a(28), 729, a(30), 2702258, 6633, 118810, 15625, 6379479. (End)

Main diagonal of A333429.

Cf. A333433.

a(n) = {my(c=0, m=0); while(cJinyuan Wang, Mar 25 2020

def a(n):
    if n == 1: return 1
    m = 0
    for c in range(1, n+1):
        m += 1
        while not (pow(n, m, m) + 1)%m == 0: m += 1
    return m
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Aug 28 2021

a(n) = A333429(n,n).

cp's OEIS Frontend

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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Mathematica

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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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A015968 Positive integers k such that k | (15^k + 1).

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A333432 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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A092067 a(n) is the smallest number m such that m > 1 and m divides n^m + 1.

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A333430 a(n) is the n-th number m that divides n^m + 1 (or 0 if m does not exist).

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