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A174844 Primes that generate three other primes when 2, 6, and 8, respectively, are subtracted from each digit of their decimal representations.

%I A174844 #13 Nov 15 2022 12:24:38
%S A174844 9898898899,889898999999,889989889889,898888889989,989899998889,
%T A174844 999988988989,988898889999899,989998888989889,98888888989989899,
%U A174844 99999998998988999,888898989989989999,888998889889898899,889888889998888999,889888898999988989,889988888998998889
%N A174844 Primes that generate three other primes when 2, 6, and 8, respectively, are subtracted from each digit of their decimal representations.
%C A174844 Subsequence of A020472. The primes generated from the subtractions are in A020469, A020458, and A020449, respectively. Final digits are necessarily 9 (here), then 7, 3, and 1. Because leading 8's are permitted in the terms here, the primes generated by subtracting 8's may have fewer digits than the others.
%H A174844 Robert Israel, <a href="/A174844/b174844.txt">Table of n, a(n) for n = 1..1188</a>
%e A174844 9898898899 is prime and so are 7676676677, 3232232233, and 1010010011, so it is a term. Although 9349, 9349-2222=7127, 9349-6666=2683, and 9349-8888=461 are four primes, 9349 is not a term as subtracting 6 or 8 from the digits 3 and 4 is not possible (no "borrowing" is permitted).
%p A174844 Res:= NULL: count:= 0:
%p A174844 for d from 2 while count < 100 do
%p A174844   v:= (10^d-1)/9;
%p A174844   for m from 1 to d do
%p A174844     if m mod 3 <> 0 and 2*d+m mod 3 <> 0 then
%p A174844       for S in combinat:-choose([$1..(d-2)],m-1) do
%p A174844         q:= 1+add(10^i,i=S);
%p A174844         if andmap(isprime, [q, 2*v+q, 6*v+q, 8*v+q]) then
%p A174844            count:= count+1; Res:= Res, 8*v+q;
%p A174844         fi
%p A174844       od;
%p A174844     fi
%p A174844   od;
%p A174844 od:
%p A174844 sort([Res]); # _Robert Israel_, Nov 14 2022
%t A174844 okQ[n_]:=Module[{idn=IntegerDigits[n]},And@@PrimeQ[FromDigits/@ {idn-2, idn-6, idn-8}]]; Select[Flatten[Table[Select[FromDigits/@ Tuples[ {8,9},n], PrimeQ],{n,18}]],okQ] (* _Harvey P. Dale_, Jul 27 2011 *)
%o A174844 (PARI) {/* Program based on that of M. F. Hasler in A020472. */
%o A174844 for(nd=1, 20, p=vector(nd, i, 10^(nd-i))~; r=(10^nd-1)/9;
%o A174844 forvec(v=vector(nd, i, [8+(i==nd), 9]), q=v*p; isprime(q) &&
%o A174844 isprime(q-2*r ) && isprime(q-6*r ) && isprime(q-8*r ) && print1(q", ")))}
%o A174844 (Python)
%o A174844 from sympy import isprime
%o A174844 from itertools import count, islice, product
%o A174844 def agen(): # generator of terms
%o A174844     for d in count(2):
%o A174844         subs = list(map(int, ["2"*d, "6"*d, "8"*d]))
%o A174844         for first in product("89", repeat=d-1):
%o A174844             t = int("".join(first) + "9")
%o A174844             if isprime(t) and all(isprime(t-s) for s in subs): yield t
%o A174844 print(list(islice(agen(), 15))) # _Michael S. Branicky_, Nov 15 2022
%Y A174844 Cf. A020472, A020469, A020458, A020449.
%K A174844 base,nonn
%O A174844 1,1
%A A174844 _Rick L. Shepherd_, Mar 30 2010