login
A062115
Numbers with no prime substring in their decimal expansion.
11
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
OFFSET
1,3
COMMENTS
This is a 10-automatic sequence, a consequence of the finitude of A071062. - Charles R Greathouse IV, Sep 27 2011
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
LINKS
Jeffrey Shallit, Minimal primes, Journal of Recreational Mathematics 30:2 (1999-2000), pp. 113-117.
FORMULA
A039997(a(n)) = 0. - Reinhard Zumkeller, Jul 16 2007
From Charles R Greathouse IV, Mar 23 2010: (Start)
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)
EXAMPLE
25 is not included because 5 is prime.
PROG
(Haskell)
a062115 n = a062115_list !! (n-1)
a062115_list = filter ((== 0) . a039997) a084984_list
-- Reinhard Zumkeller, Jan 31 2012
(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)
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, May 02 2023
(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)))
print(list(islice(agen(), 100))) # Michael S. Branicky, May 02 2023
CROSSREFS
Subsequence of A084984. [Arkadiusz Wesolowski, Jul 05 2011]
Cf. A071062.
Cf. A163753 (complement).
Sequence in context: A050695 A035139 A316227 * A141613 A072362 A278948
KEYWORD
base,easy,nonn
AUTHOR
Erich Friedman, Jun 28 2001
EXTENSIONS
Offset corrected by Arkadiusz Wesolowski, Jul 27 2011
STATUS
approved