A239019 Numbers which are not primitive words over the alphabet {0,...,9} (when written in base 10).

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4444, 4545, 4646, 4747, 4848, 4949

Offset: 1

Author

M. F. Hasler, Mar 08 2014

Keywords

• nonn
• base

Comments

A word is primitive iff it is not a power, i.e., repetition, of a subword. The only non-primitive words with a prime number of letters (here: digits) are the repdigit numbers. Thus, the first nontrivial terms of this sequence are 1010,1212,...

This sequence does *not* contain all non-primitive words over the alphabet {0,...,9}, namely, it excludes those which would be numbers with leading zeros: 00,000,0000,0101,0202,...

Lists of non-primitive words over a sub-alphabet of {1...9}, like A213972, A213973, A213974, A239018, ... are given as intersection of this with the set of all words in that alphabet, e.g., A007931, A032810, A032917, A007932, ...

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

F:= proc(d) local p,R,q;
  R:= {seq(x*(10^d-1)/9, x=1..9)};
  for p in numtheory:-factorset(d) minus {d} do
    q:= d/p;
    R:= R union {seq(x*(10^d-1)/(10^q-1),x=10^(q-1)..10^q-1)};
  od:
  sort(convert(R,list))
end proc:
[seq(op(F(i)),i=2..4)]; # Robert Israel, Nov 14 2017

PARI

is_A239019(n)=fordiv(#n=digits(n),L,L<#n && n==concat(Col(vector(#n/L,i,1)~*vecextract(n,2^L-1))~)&&return(1))