A257291 Number of states in minimal DFA accepting base-2 representation of first n prime numbers.

4, 4, 5, 5, 6, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 16, 17, 17, 19, 20, 21, 21, 21, 22, 23, 23, 24, 24, 24, 25, 26, 27, 28, 28, 29, 29, 31, 32, 33, 35, 36, 36, 37, 38, 38, 38, 39, 40, 41, 40, 41, 41, 42, 43, 44, 44, 44, 45, 46, 46, 47, 47, 48, 48, 49, 49, 49, 51, 52, 52, 54, 55, 55, 56, 56, 57, 58, 58

Offset: 1

Author

Jeffrey Shallit, Apr 21 2015

Keywords

• nonn

Comments

By "DFA" we mean deterministic finite automaton, which must be "complete" (that is, a transition must exist for every state). So the minimal DFA for n = 1 corresponds to a DFA that accepts the string "10" and no other. Four states are required since a "dead state" is also needed.

Examples

			From _Kevin Ryde_, Jun 02 2020: (Start)
For n=3, the minimum DFA comprises a(3) = 5 states:
                            +------------------------+
     start                1 |                        v
  +-----------+  1   +--------+  0   +=====+  1   +=====+
  | 10,11,101 | ---> | 0,1,01 | ---> | e,1 | ---> |  e  |
  +-----------+      +--------+      +=====+      +=====+
        | 0                             | 0          | 0,1
        |                               |            v
        |                               |        +------+
        +-------------------------------+------> | dead |
                                                 +------+
                                                   ^ | 0,1
                                                   +-+
Each state is a set of bit strings wanted.  The start state is primes 2,3,5 in binary.  Each "1" transition takes strings starting 1 and removes that 1.  Each 0 transition similarly.  "e" is the empty string.  Each state containing "e" is accepting because it's the end of one of the original primes.  "Dead" is the set of no strings and is a non-accepting sink.  Input strings too long or not a prefix of one of the desired primes end up at dead.
(End)
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• Kevin Ryde, Perl and Perl+Foma code to generate the b-file

Crossrefs

Cf. A257371.

Programs

PARI

a(n) = {
  my(m=Map(),q=List([apply(p->Vecsmall(binary(p)),primes(n))]));
  while(#q, my(s=q[#q]);listpop(q);
    if(!mapisdefined(m,s), mapput(m,s,1);
      for(i=0,1, listput(q,apply(v->v[^1],
                             select(v->#v&&v[1]==i, s))))));
  #m; }  \\ Kevin Ryde, Jun 02 2020

Python

from sympy import prime, primerange
def a(n):
    m = dict()
    q = [tuple(bin(p)[2:] for p in primerange(1, prime(n)+1))]
    while len(q) > 0:
        s = q.pop()
        if s not in m:
            m[s] = 1
            for i in "01":
                q.append(tuple(v[1:] for v in s if len(v) and v[0]==i))
    return len(m)
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Jul 04 2022 after Kevin Ryde

Extensions

a(26)-a(78) from Kevin Ryde, Jun 02 2020