A257291 - OEIS

%I #47 Jul 05 2022 02:31:19

%S 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,

%T 23,24,24,24,25,26,27,28,28,29,29,31,32,33,35,36,36,37,38,38,38,39,40,

%U 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

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

%C 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.

%H Kevin Ryde, <a href="/A257291/b257291.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevin Ryde, <a href="/A257291/a257291.pl.txt">Perl and Perl+Foma code to generate the b-file</a>

%e From _Kevin Ryde_, Jun 02 2020: (Start)

%e For n=3, the minimum DFA comprises a(3) = 5 states:

%e                             +------------------------+

%e      start                1 |                        v

%e   +-----------+  1   +--------+  0   +=====+  1   +=====+

%e   | 10,11,101 | ---> | 0,1,01 | ---> | e,1 | ---> |  e  |

%e   +-----------+      +--------+      +=====+      +=====+

%e         | 0                             | 0          | 0,1

%e         |                               |            v

%e         |                               |        +------+

%e         +-------------------------------+------> | dead |

%e                                                  +------+

%e                                                    ^ | 0,1

%e                                                    +-+

%e 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.

%e (End)

%o (PARI) a(n) = {

%o   my(m=Map(),q=List([apply(p->Vecsmall(binary(p)),primes(n))]));

%o   while(#q, my(s=q[#q]);listpop(q);

%o     if(!mapisdefined(m,s), mapput(m,s,1);

%o       for(i=0,1, listput(q,apply(v->v[^1],

%o                              select(v->#v&&v[1]==i, s))))));

%o   #m; }  \\ _Kevin Ryde_, Jun 02 2020

%o (Python)

%o from sympy import prime, primerange

%o def a(n):

%o     m = dict()

%o     q = [tuple(bin(p)[2:] for p in primerange(1, prime(n)+1))]

%o     while len(q) > 0:

%o         s = q.pop()

%o         if s not in m:

%o             m[s] = 1

%o             for i in "01":

%o                 q.append(tuple(v[1:] for v in s if len(v) and v[0]==i))

%o     return len(m)

%o print([a(n) for n in range(1, 80)]) # _Michael S. Branicky_, Jul 04 2022 after _Kevin Ryde_

%Y Cf. A257371.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Apr 21 2015

%E a(26)-a(78) from _Kevin Ryde_, Jun 02 2020