A333433 -id:A333433 - cp's OEIS frontend

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.

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 *)

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).

A333502 a(n) is the n-th number m such that m^2 divides n^m - 1 (or 0 if m does not exist).

Original entry on oeis.org

1, 0, 4, 903, 12, 776119592182705, 12, 42931441, 136, 27486820443, 60, 107342336783, 84

Offset: 1

Seiichi Manyama, Mar 24 2020

Main diagonal of A333500.

Cf. A127106, A127100, A128405, A333433.

{a(n) = if(n==2, 0, my(cnt=0, k=0); while(cnt

a(n) = A333500(n,n).

cp's OEIS Frontend

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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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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A333502 a(n) is the n-th number m such that m^2 divides n^m - 1 (or 0 if m does not exist).

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