OFFSET
1,1
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 63 terms from Rodrigo de O. Leite)
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rodrigo de O. Leite, Dec 10 2016
STATUS
approved