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A213972 List of imprimitive words over the alphabet {1,2}. 5
11, 22, 111, 222, 1111, 1212, 2121, 2222, 11111, 22222, 111111, 112112, 121121, 121212, 122122, 211211, 212121, 212212, 221221, 222222, 1111111, 2222222, 11111111, 11121112, 11211121, 11221122, 12111211, 12121212, 12211221, 12221222, 21112111, 21122112, 21212121 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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 A213973 by replacing all digits 3 by 2 and from A213974 by replacing digits 2 by 1 and digits 3 by 2. - 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.
LINKS
FORMULA
A213972 = A007931 intersect A239018. - M. F. Hasler, Mar 10 2014
MAPLE
P:= proc(d) option remember; 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;
sort(A);
end proc:
IP:= proc(d)
sort([seq(seq(s*(10^d-1)/(10^m-1), s = P(m)), m=numtheory:-divisors(d) minus {d})]);
end proc:
seq(op(IP(d)), d=1..10); # Robert Israel, Mar 24 2017
MATHEMATICA
j[w_, k_] := FromDigits /@ (Flatten[Table[#, {k}]] & /@ w); Flatten@ Table[ Union@ Flatten[ j[Tuples [{1, 2}, #], n/#] & /@ Most@ Divisors@ n], {n, 9}] (* Giovanni Resta, Mar 24 2017 *)
PROG
(PARI) for(n=1, 10, p=vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 2]), is_A239017(m=d*p)||print1(m", "))) \\ M. F. Hasler, Mar 10 2014
CROSSREFS
See A239018 for the analog over the alphabet {1,2,3}.
Sequence in context: A143964 A034708 A091784 * A061852 A083511 A067894
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Jun 30 2012
EXTENSIONS
More terms from M. F. Hasler, Mar 10 2014
STATUS
approved

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Last modified May 21 16:27 EDT 2024. Contains 372738 sequences. (Running on oeis4.)