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A062115
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Numbers with no prime substring in their decimal expansion.
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11
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0, 1, 4, 6, 8, 9, 10, 14, 16, 18, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 81, 84, 86, 88, 90, 91, 94, 96, 98, 99, 100, 104, 106, 108, 140, 144, 146, 148, 160, 164, 166, 168, 169, 180, 184, 186, 188, 400, 404, 406, 408, 440, 444, 446, 448, 460, 464, 466
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OFFSET
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1,3
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COMMENTS
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Subsequence of A202259 (right-truncatable nonprimes). Supersequence of A202262 (composite numbers in which all substrings are composite), A202265 (nonprime numbers in which all substrings and reversal substrings are nonprimes). - Jaroslav Krizek, Jan 28 2012
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LINKS
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Jeffrey Shallit, Minimal primes, Journal of Recreational Mathematics 30:2 (1999-2000), pp. 113-117.
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FORMULA
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a(n) = O(n^(log_4 10)) = O(n^1.661) because numbers containing only 0,4,6,8 are in this sequence.
a(n) = Omega(n^(log_13637 1000000)) = Omega(n^1.451) for similar reasons; this bound can be increased by considering longer sequences of digits. (End)
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EXAMPLE
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25 is not included because 5 is prime.
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PROG
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(Haskell)
a062115 n = a062115_list !! (n-1)
a062115_list = filter ((== 0) . a039997) a084984_list
(Python)
from sympy import isprime
def ok(n):
s = str(n)
ss = (int(s[i:j]) for i in range(len(s)) for j in range(i+1, len(s)+1))
return not any(isprime(k) for k in ss)
(Python) # faster for initial segment of sequence; uses ok, import above
from itertools import chain, count, islice, product
def agen(): # generator of terms
yield from chain((0, ), (int(t) for t in (f+"".join(r) for d in count(1) for f in "14689" for r in product("014689", repeat=d-1)) if ok(t)))
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CROSSREFS
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KEYWORD
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base,easy,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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