A127092
Numbers k such that k^2 divides 11^k - 1.
Original entry on oeis.org
1, 2, 4, 5, 6, 10, 12, 20, 30, 42, 60, 84, 114, 156, 210, 222, 228, 244, 420, 444, 570, 732, 780, 798, 930, 1092, 1110, 1140, 1220, 1554, 1596, 1806, 1860, 2220, 2436, 2964, 3108, 3612, 3660, 3990, 4218, 5124, 5460, 5772, 6510, 7770, 7980, 8268, 8436, 9030
Offset: 1
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Select[Range[15000], IntegerQ[(PowerMod[11, #, #^2 ]-1)/#^2 ]&]
Join[{1},Select[Range[9100],PowerMod[11,#,#^2]==1&]] (* Harvey P. Dale, Dec 30 2018 *)
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for(k=1, 1e4, if(Mod(11, k^2)^k==1, print1(k", "))) \\ Seiichi Manyama, Mar 25 2020
A128399
Numbers k such that k^2 divides 19^k-1.
Original entry on oeis.org
1, 2, 3, 4, 6, 9, 10, 12, 18, 20, 30, 36, 42, 60, 84, 90, 110, 126, 156, 180, 210, 220, 252, 294, 330, 381, 420, 468, 588, 630, 660, 724, 762, 780, 882, 930, 990, 1092, 1143, 1260, 1332, 1470, 1510, 1524, 1764, 1806, 1830, 1860, 1980, 2028, 2058, 2172, 2286, 2310
Offset: 1
Cf.
A127103,
A127104,
A127105,
A127106,
A127107,
A127102,
A127101,
A127100,
A127092,
A128405,
A128393,
A128394,
A128395,
A128396,
A128397,
A128398,
A128400,
A128401,
A128402,
A128403,
A128404.
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Join[{1},Select[Range[3000],PowerMod[19,#,#^2]==1&]] (* Harvey P. Dale, Oct 24 2017 *)
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for(k=1, 1e4, if(Mod(19, k^2)^k==1, print1(k", "))) \\ Seiichi Manyama, Mar 25 2020
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
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, ...
Columns k=1-20 give:
A000027,
A063524,
A067945,
A014945,
A067946,
A014946,
A067947,
A014949,
A068382,
A014950,
A068383,
A014951,
A116621,
A177805,
A014957,
A177807,
A128358,
A333506,
A128360.
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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
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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 *)
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
Showing 1-4 of 4 results.
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