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

9898898899, 889898999999, 889989889889, 898888889989, 989899998889, 999988988989, 988898889999899, 989998888989889, 98888888989989899, 99999998998988999, 888898989989989999, 888998889889898899, 889888889998888999, 889888898999988989, 889988888998998889

Offset: 1

Author

Rick L. Shepherd, Mar 30 2010

Keywords

• base
• nonn

Comments

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.

Examples

			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).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1188

Crossrefs

Cf. A020472, A020469, A020458, A020449.

Programs

Maple

Res:= NULL: count:= 0:
for d from 2 while count < 100 do
  v:= (10^d-1)/9;
  for m from 1 to d do
    if m mod 3 <> 0 and 2*d+m mod 3 <> 0 then
      for S in combinat:-choose([$1..(d-2)],m-1) do
        q:= 1+add(10^i,i=S);
        if andmap(isprime, [q, 2*v+q, 6*v+q, 8*v+q]) then
           count:= count+1; Res:= Res, 8*v+q;
        fi
      od;
    fi
  od;
od:
sort([Res]); # Robert Israel, Nov 14 2022

Mathematica

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 *)

PARI

{/* Program based on that of M. F. Hasler in A020472. */
for(nd=1, 20, p=vector(nd, i, 10^(nd-i))~; r=(10^nd-1)/9;
forvec(v=vector(nd, i, [8+(i==nd), 9]), q=v*p; isprime(q) &&
isprime(q-2*r ) && isprime(q-6*r ) && isprime(q-8*r ) && print1(q", ")))}

Python

from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    for d in count(2):
        subs = list(map(int, ["2"*d, "6"*d, "8"*d]))
        for first in product("89", repeat=d-1):
            t = int("".join(first) + "9")
            if isprime(t) and all(isprime(t-s) for s in subs): yield t
print(list(islice(agen(), 15))) # Michael S. Branicky, Nov 15 2022