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%I #48 Oct 27 2023 18:28:26
%S 89,449,499,4649,4969,6469,6869,6949,8669,8699,8849,9649,9949,44699,
%T 46649,48649,48869,49669,64849,84869,86969,88469,94849,94949,98869,
%U 99469,444469,444869,446969,466649,468869,469849,469969,494699,496669,496849,498469,644669
%N Primes whose proper substrings of consecutive digits are all composite.
%C All digits are composite. Each term ends with the digit '9'. Since each term is prime, it never serves as the suffix of any subsequent term; e.g., no term beyond 89 ends with the digits '89', so the only remaining allowed two-digit endings are '49', '69', and '99'; no terms beyond 449 and 499 end with '449' or '499' (and '899' is ruled out because of 89), so the only remaining allowed three-digit endings are '469', '649', '669', '699', '849', '869', '949', '969', and '999' (and each of these appears as the ending of at least one four-digit term, except '999', which doesn't appear as the ending of any term until a(75) = 4696999). - _Jon E. Schoenfield_, Dec 10 2016
%C Number of terms < 10^k, k=1,2,3,...: 0, 1, 2, 10, 13, 38, 66, 197, 410, 1053, 2542, 7159, 18182, 49388, ..., . _Robert G. Wilson v_, Jan 15 2017
%H Alois P. Heinz, <a href="/A279366/b279366.txt">Table of n, a(n) for n = 1..10000</a> (first 63 terms from Rodrigo de O. Leite)
%e 44699 is in the sequence because 4, 6, 9, 44, 46, 69, 99, 446, 469, 669, 4469 and 4699 are composite numbers. However, 846499 is not included because 4649 is prime.
%t Select[Prime@ Range[5, 10^5], Function[n, Times @@ Boole@ Map[CompositeQ, Flatten@ Map[FromDigits /@ Partition[n, #, 1] &, Range[Length@ n - 1]]] == 1]@ IntegerDigits@ # &] (* _Michael De Vlieger_, Dec 10 2016 *)
%t Select[Flatten[Table[FromDigits/@Tuples[{4,6,8,9},d],{d,6}]],PrimeQ[#]&&AllTrue[ FromDigits /@ Union[Flatten[Table[Partition[IntegerDigits[#],n,1],{n,IntegerLength[#]-1}],1]], CompositeQ]&] (* _Harvey P. Dale_, Jul 15 2023 *)
%o (Python)
%o from sympy import isprime
%o from itertools import count, islice, product
%o def ok(n):
%o s = str(n)
%o if set(s) & {"1", "2", "3", "5", "7"} or not isprime(n): return False
%o ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
%o return not any(isprime(int(ss)) for ss in ss2)
%o def agen():
%o for d in count(2):
%o for p in product("4689", repeat=d-1):
%o t = int("".join(p)+"9")
%o if ok(t): yield t
%o print(list(islice(agen(), 38))) # _Michael S. Branicky_, Oct 07 2022
%Y Subsequence of A051416.
%Y Cf. A033274, A061372, A254754.
%K nonn,base
%O 1,1
%A _Rodrigo de O. Leite_, Dec 10 2016
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