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A257371
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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
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2
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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