cp's OEIS Frontend

A213972 List of imprimitive words over the alphabet {1,2}.

%I A213972 #21 Mar 24 2017 22:03:03
%S A213972 11,22,111,222,1111,1212,2121,2222,11111,22222,111111,112112,121121,
%T A213972 121212,122122,211211,212121,212212,221221,222222,1111111,2222222,
%U A213972 11111111,11121112,11211121,11221122,12111211,12121212,12211221,12221222,21112111,21122112,21212121
%N A213972 List of imprimitive words over the alphabet {1,2}.
%C A213972 A word w is primitive if it cannot be written as u^k with k>1; otherwise it is imprimitive.
%C A213972 The {0,1} version of this sequence is
%C A213972 00, 11, 000, 111, 0000, 0101, 1010, 1111, 00000, 11111, 000000, 001001, 010010, 010101, 011011, 100100, 101010, 101101, 110110, 111111
%C A213972 but this cannot be included as a sequence in the OEIS since it contains nonzero "numbers" beginning with 0.
%C A213972 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
%D A213972 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.
%H A213972 Robert Israel, <a href="/A213972/b213972.txt">Table of n, a(n) for n = 1..10000</a>
%F A213972 A213972 = A007931 intersect A239018. - _M. F. Hasler_, Mar 10 2014
%p A213972 P:= proc(d) option remember;local m,A;
%p A213972     A:= map(t -> (10^d-1)/9 + add(10^s, s = t), combinat:-powerset([$0..d-1]));
%p A213972     for m in numtheory:-divisors(d) minus {d} do
%p A213972       A:= remove(t -> t = (t mod 10^m)*(10^d-1)/(10^m-1), A);
%p A213972     od;
%p A213972     sort(A);
%p A213972 end proc:
%p A213972 IP:= proc(d)
%p A213972    sort([seq(seq(s*(10^d-1)/(10^m-1), s = P(m)), m=numtheory:-divisors(d) minus {d})]);
%p A213972 end proc:
%p A213972 seq(op(IP(d)), d=1..10); # _Robert Israel_, Mar 24 2017
%t A213972 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 *)
%o A213972 (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
%Y A213972 Cf. A213969-A213974.
%Y A213972 See A239018 for the analog over the alphabet {1,2,3}.
%K A213972 nonn,base
%O A213972 1,1
%A A213972 _N. J. A. Sloane_, Jun 30 2012
%E A213972 More terms from _M. F. Hasler_, Mar 10 2014