A239018 Non-primitive words on {1,2,3}.

11, 22, 33, 111, 222, 333, 1111, 1212, 1313, 2121, 2222, 2323, 3131, 3232, 3333, 11111, 22222, 33333, 111111, 112112, 113113, 121121, 121212, 122122, 123123, 131131, 131313, 132132, 133133, 211211, 212121, 212212, 213213, 221221, 222222, 223223, 231231, 232232, 232323, 233233, 311311, 312312, 313131, 313313

Offset: 1

Author

M. F. Hasler, Mar 08 2014

Keywords

• nonn

Comments

A word is non-primitive if it is a nontrivial power (i.e., repetition) of a subword. Therefore, for a prime number of digits, only the repdigit numbers are primitive. For words with 6 letters, there is also 112^2,113^2,121^2,12^3,... where w^n means n concatenations of w.

Lyndon words on {1,2,3}, A102660, are the terms in A007932 which are primitive (i.e., in the complement A239017 of this sequence) and not larger than any of their rotation, i.e., in A239016.

This is the complement of A239017 in A007932.

This is for {1,2,3} what A213972 is for {1,2} (and A213973 for {1,3}, A213974 for {2,3}).

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10239 (all terms with <= 16 digits)

Crossrefs

Cf. A102659, A213969 - A213974.

Cf. A007932, A239017.

Programs

PARI

for(n=1,7,p=vector(n,i,10^(n-i))~;forvec(d=vector(n,i,[1,3]),is_A239017(m=d*p)||print1(m",")))

Python

from sympy import divisors
from itertools import product
def agentod(maxd):
    for d in range(2, maxd+1):
        divs, alld = divisors(d)[:-1], set()
        for div in divs:
            for t in product("123", repeat=div):
                alld.add(int("".join(t*(d//div))))
        yield from sorted(alld)
print([an for an in agentod(6)]) # Michael S. Branicky, Nov 22 2021