A111488 Primes having only {0, 1, 3, 6} as digits.

3, 11, 13, 31, 61, 101, 103, 113, 131, 163, 311, 313, 331, 601, 613, 631, 661, 1013, 1031, 1033, 1061, 1063, 1103, 1163, 1301, 1303, 1361, 1601, 1613, 1663, 3001, 3011, 3061, 3163, 3301, 3313, 3331, 3361, 3613, 3631, 6011, 6101, 6113, 6131, 6133, 6163

Offset: 1

Author

Jonathan Vos Post, Nov 15 2005

Keywords

• base
• easy
• nonn

Comments

Includes all repunit primes (A004022). Conjecture: an infinite sequence. Note twin primes: (11, 13), (101, 103), (311, 313), (1031, 1033), (1061, 1063), (1301, 1303), (6131, 6133), (10301, 10303), (10331, 10333), (13001, 13003).

In other words, primes with digits in the set {0,1,3,6}. - M. F. Hasler, Jul 25 2015

The number of 1's in the representation must be either 1 or 2 (mod 3), because otherwise the number would be divisible by 3 (and therefore composite). The only exception is the 3 itself. This excludes basically members of A038603. - R. J. Mathar, Jul 25 2015

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index to entries for primes with digits in a given set

Crossrefs

Cf. A000040, A000217, A004022, A038603.

Cf. also A020450 - A020472, A036953, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349.

Programs

Maple

f:= proc(x) local L,p;
  L:= subs([3=6,2=3],convert(x,base,4));
  p:= add(L[i]*10^(i-1),i=1..nops(L));
  if isprime(p) then p fi
end proc:
map(f, [$1..4^4]); # Robert Israel, Dec 18 2018

Mathematica

Select[Prime@ Range@ 1000, SubsetQ[{0, 1, 3, 6}, IntegerDigits@ #] &] (* Michael De Vlieger, Jul 25 2015 *)

PARI

A111488={(n, show=0, L=[0,1,3,6])->my(t); for(d=1,1e9,u=vector(d, i, 10^(d-i))~; forvec(v=vector(d,i,[1+(i==1&&!L[1]), #L]), ispseudoprime(t=vector(d, i, L[v[i]])*u)||next; show&print1(t", "); n--||return(t)))} \\ M. F. Hasler, Jul 25 2015

Extensions

Corrected by Ray Chandler, Nov 19 2005

Name changed by Sean A. Irvine, Jul 21 2025