A279366 Primes whose proper substrings of consecutive digits are all composite.

89, 449, 499, 4649, 4969, 6469, 6869, 6949, 8669, 8699, 8849, 9649, 9949, 44699, 46649, 48649, 48869, 49669, 64849, 84869, 86969, 88469, 94849, 94949, 98869, 99469, 444469, 444869, 446969, 466649, 468869, 469849, 469969, 494699, 496669, 496849, 498469, 644669

Offset: 1

Author

Rodrigo de O. Leite, Dec 10 2016

Keywords

• nonn
• base

Comments

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

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 63 terms from Rodrigo de O. Leite)

Crossrefs

Subsequence of A051416.

Cf. A033274, A061372, A254754.

Programs

Mathematica

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 *)
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 *)

Python

from sympy import isprime
from itertools import count, islice, product
def ok(n):
    s = str(n)
    if set(s) & {"1", "2", "3", "5", "7"} or not isprime(n): return False
    ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
    return not any(isprime(int(ss)) for ss in ss2)
def agen():
    for d in count(2):
        for p in product("4689", repeat=d-1):
            t = int("".join(p)+"9")
            if ok(t): yield t
print(list(islice(agen(), 38))) # Michael S. Branicky, Oct 07 2022