A003060
Smallest number with reciprocal of period length n in decimal (base 10).
Original entry on oeis.org
1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, 21649, 707, 53, 2629, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 511, 21401, 583, 243, 29, 3191, 211, 2791, 353, 67, 103, 71, 1919, 2028119, 909090909090909091
Offset: 0
- J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
- "Cycle lengths of reciprocals", Popular Computing (Calabasas, CA), Vol. 1 (No. 4, Jul 1973), pp. 12-14.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Smallest primitive divisors of b^n-1:
A212953 (b=2),
A218356 (b=3),
A218357 (b=5),
A218358 (b=7), this sequence (b=10),
A218359 (b=11),
A218360 (b=13),
A218361 (b=17),
A218362 (b=19),
A218363 (b=23),
A218364 (b=29).
-
a[n_] := First[ Select[ Divisors[10^n - 1], MultiplicativeOrder[10, #] == n &, 1]]; a[0] = 1; a[1] = 3; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 13 2012, after T. D. Noe *)
-
apply( {A003060(n)=!fordiv(10^n-!!n, d, d>1 && znorder(Mod(10,d))==n && return(d))}, [0..50]) \\ M. F. Hasler, Jun 28 2022
b-file truncated at uncertain term a(439) by
Max Alekseyev, Apr 30 2022
A213224
Minimal order A(n,k) of degree-n irreducible polynomials over GF(prime(k)); square array A(n,k), n>=1, k>=1, read by antidiagonals.
Original entry on oeis.org
1, 1, 3, 1, 4, 7, 1, 3, 13, 5, 1, 4, 31, 5, 31, 1, 3, 9, 13, 11, 9, 1, 7, 7, 5, 11, 7, 127, 1, 3, 9, 16, 2801, 7, 1093, 17, 1, 4, 307, 5, 25, 36, 19531, 32, 73, 1, 3, 27, 5, 30941, 9, 29, 32, 757, 11, 1, 3, 7, 16, 88741, 63, 43, 64, 19, 44, 23
Offset: 1
A(4,1) = 5: The degree-4 irreducible polynomials p over GF(prime(1)) = GF(2) are 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. Their orders (i.e., the smallest integer e for which p divides x^e+1) are 5, 15, 15, and the minimal order is 5. (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2).
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
3, 4, 3, 4, 3, 7, 3, 4, ...
7, 13, 31, 9, 7, 9, 307, 27, ...
5, 5, 13, 5, 16, 5, 5, 16, ...
31, 11, 11, 2801, 25, 30941, 88741, 151, ...
9, 7, 7, 36, 9, 63, 7, 7, ...
127, 1093, 19531, 29, 43, 5229043, 25646167, 701, ...
17, 32, 32, 64, 32, 32, 128, 17, ...
-
with(numtheory):
M:= proc(n, i) option remember;
divisors(ithprime(i)^n-1) minus U(n-1, i)
end:
U:= proc(n, i) option remember;
`if`(n=0, {}, M(n, i) union U(n-1, i))
end:
A:= (n, k)-> min(M(n, k)[]):
seq(seq(A(n, d+1-n), n=1..d), d=1..14);
-
M[n_, i_] := M[n, i] = Divisors[Prime[i]^n - 1] ~Complement~ U[n-1, i]; U[n_, i_] := U[n, i] = If[n == 0, {}, M[n, i] ~Union~ U[n-1, i]]; A[n_, k_] := Min[M[n, k]]; Table[Table[A[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)
A218341
Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(29) listed in ascending order.
Original entry on oeis.org
1, 2, 4, 7, 14, 28, 3, 5, 6, 8, 10, 12, 15, 20, 21, 24, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840, 13, 26, 52, 67, 91, 134, 182, 268, 364, 469, 871, 938, 1742, 1876, 3484, 6097, 12194, 24388, 16, 48, 80, 112, 240, 336, 421, 560
Offset: 1
Triangle begins:
1, 2, 4, 7, 14, 28;
3, 5, 6, 8, 10, 12, 15, ...
13, 26, 52, 67, 91, 134, 182, ...
16, 48, 80, 112, 240, 336, 421, ...
732541, 1465082, 2930164, 5127787, 10255574, 20511148;
...
-
with(numtheory):
M:= proc(n) M(n):= divisors(29^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);
-
M[n_] := M[n] = Divisors[29^n-1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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