A213969 List of primitive words over the alphabet {1,2}.

1, 2, 12, 21, 112, 121, 122, 211, 212, 221, 1112, 1121, 1122, 1211, 1221, 1222, 2111, 2112, 2122, 2211, 2212, 2221, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21211, 21212, 21221, 21222, 22111, 22112, 22121, 22122, 22211, 22212, 22221

Offset

1,2

Comments

A word w is primitive if it cannot be written as u^k with k>1; otherwise it is imprimitive.

The {0,1} version of this sequence is

0, 1, 01, 10, 001, 010, 011, 100, 101, 110, 0001, 0010, 0011, 0100, 0110, 0111, 1000, 1001, 1011, 1100, 1101, 1110, 00001, 00010, 00011, 00100, 00101, 00110, 00111, 01000, 01001, 01010, 01011, 01100, 01101, 01110, 01111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, ...,

but this cannot be included as a sequence in the OEIS since it contains nonzero "numbers" beginning with 0.

Lyndon words on {1,2}, A102659, are the numbers in this sequence which are also not larger than any of their rotations, i.e., in A239016. - M. F. Hasler, Mar 08 2014

References

A. de Luca and S. Varricchio, Finiteness and Regularity in Semigroups and Formal Languages, Monographs in Theoretical Computer Science, Springer-Verlag, Berlin, 1999. See p. 10.

Links

Robert Israel, Table of n, a(n) for n = 1..16222 (all terms with up to 13 digits)

Maple

P:= proc(d) local m, A;
    A:= map(t -> (10^d-1)/9 + add(10^s, s = t), combinat:-powerset([$0..d-1]));
    for m in numtheory:-divisors(d) minus {d} do
      A:= remove(t -> t = (t mod 10^m)*(10^d-1)/(10^m-1), A);
    od;
    op(sort(A));
end proc:
seq(P(d), d=1..6); # Robert Israel, Mar 24 2017

Mathematica

j[w_, k_] := FromDigits /@ (Flatten[Table[#, {k}]] & /@ w); L[n_] := Complement[ FromDigits /@ Tuples[{1, 2}, n], Union[ Flatten[( j[Tuples[{1, 2}, #1], n/#1] &) /@ Most[ Divisors[n]]]]]; Flatten@ Array[L, 5] (* Giovanni Resta, Mar 24 2017 *)

Prog

(PARI) is_A213969(n)={fordiv(#n=digits(n), L, L<#n&&n==concat(Col(vector(#n/L, i, 1)~*vecextract(n, 2^L-1))~)&&return); !setminus(Set(n), [1, 2])}
for(n=1, 5, p=vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 2]), is_A213969(m=d*p)&&print1(m", "))) \\ M. F. Hasler, Mar 08 2014

Crossrefs

Cf. A213969-A213974.

Sequence in context: A053890 A053896 A155890 * A199986 A336528 A077410

Adjacent sequences: A213966 A213967 A213968 * A213970 A213971 A213972

Keyword

nonn,base

Author

N. J. A. Sloane, Jun 30 2012

Status

approved