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A257371 Number of states in minimal DFA for P_n^*, where P_n is the set of base-2 representations of the first n prime numbers 2
3, 3, 5, 5, 5, 7, 8, 8, 8, 8, 8, 9, 12, 12, 12, 15, 15, 15, 17, 17, 18, 18, 18, 19, 29, 29, 31, 31, 31, 31, 31, 33, 34, 34, 35, 35, 35, 37, 37, 37, 37, 37, 37, 38, 41, 40, 41, 41, 49, 49, 49, 49, 49, 49, 51, 52, 53, 53, 55, 59, 59, 59, 62, 62, 63, 63, 65, 65 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Here by "DFA" we mean "deterministic finite automaton", and by P_n^* we mean the set of all binary strings that can be formed as the concatenation of 0 or more of the base-2 representations of the first n primes.  The DFA must be "complete" (there must be a transition for each state and symbol) and hence includes any "dead state".

P_n^* is identical to P_{n-1}^* when prime(n) in binary is the concatenation of smaller primes, since it adds nothing to the pattern.  So a(n) = a(n-1) when n in A090423, though various other a(n) = a(m) occur too when different state machines happen to have the same state count. - Kevin Ryde, Dec 26 2020

LINKS

Kevin Ryde, Table of n, a(n) for n = 1..10000

Kevin Ryde, Perl program running Foma (or others) to make a b-file

EXAMPLE

From Kevin Ryde, Dec 26 2020: (Start)

For n=3, primes 2,3,5 in binary are 10,11,101 and the regular expression is P_3^* = (10|11|101)*.  State machine manipulations, or some thought about these bits, gives the following minimum DFA comprising a(3) = 5 states.  States A,C,D are accepting.  States B and dead are non-accepting.  Dead is reached when some initial part of the input string shows the whole will not match, no matter what follows.

  start

  +===+  1   +---+  0   +===+  1   +===+

  | A | ---> | B | ---> | C | ---> | D |

  +===+ <--- +---+      +===+ <--- +===+

    |0    1               |0    0   ^ |1

    |       +------+      |         +-+

    +-----> | dead | <----+

            +------+

              ^ | 0,1

              +-+

(End)

CROSSREFS

Cf. A004676, A090423, A257291.

Sequence in context: A123313 A131507 A203998 * A075260 A054847 A335266

Adjacent sequences:  A257368 A257369 A257370 * A257372 A257373 A257374

KEYWORD

nonn

AUTHOR

Jeffrey Shallit, Apr 21 2015

EXTENSIONS

a(26)-a(68) from Kevin Ryde, Dec 27 2020

STATUS

approved

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Last modified June 24 18:38 EDT 2021. Contains 345419 sequences. (Running on oeis4.)