A213973 List of imprimitive words over the alphabet {1,3}.

11, 33, 111, 333, 1111, 1313, 3131, 3333, 11111, 33333, 111111, 113113, 131131, 131313, 133133, 311311, 313131, 313313, 331331, 333333, 1111111, 3333333, 11111111, 11131113, 11311131, 11331133, 13111311, 13131313, 13311331, 13331333, 31113111, 31133113, 31313131

Offset: 1

Author

N. J. A. Sloane, Jun 30 2012

Keywords

• nonn
• base

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

00, 11, 000, 111, 0000, 0101, 1010, 1111, 00000, 11111, 000000, 001001, 010010, 010101, 011011, 100100, 101010, 101101, 110110, 111111

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

This sequence results from A213972 by replacing all digits 2 by 3, and from A213974 by replacing all digits 2 by 1. - M. F. Hasler, Mar 10 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.

Crossrefs

Cf. A213969-A213974.

Programs

PARI

for(n=1, 8, p=2*vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 3]/2), is_A239017(m=d*p)||print1(m", ")))

Formula

A213973 = A032917 intersect A239018. - M. F. Hasler, Mar 10 2014

Extensions

More terms from M. F. Hasler, Mar 10 2014