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A213972
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List of imprimitive words over the alphabet {1,2}.
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5
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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See A239018 for the analog over the alphabet {1,2,3}.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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