A134996 Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes.

2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081, 188011, 188801, 1008001, 1022201, 1028011, 1055501, 1058011, 1082801, 1085801, 1088081, 1108201, 1108501, 1110881, 1120121, 1120211

Offset: 1

Author

Mike Keith

Keywords

• nonn
• base
• nice

Comments

The digits of a(n) are restricted to 0, 1, 2, 5, 8. - Ivan N. Ianakiev, Oct 08 2015

The first term containing all the possible digits is 108225151. There are 2958 such terms up to 10^12, the last one in this range being 188885250551. - Giovanni Resta, Oct 08 2015

Examples

			120121 is such a number because 120121, 121021 (upside down), 151051 (mirror) and 150151 are all prime. (This is the smallest one in which all four numbers are distinct.)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..7174
• C. K. Caldwell, The Prime Glossary, Dihedral Prime
• Eric Weisstein's World of Mathematics, Dihedral Prime.

Crossrefs

Cf. A134997, A134998, A048661.

Programs

Mathematica

lst1={2,5};
startQ[n_]:=First[IntegerDigits[n]]==1;
subQ[n_]:=Module[{lst={0,1,2,5,8}},SubsetQ[lst,Union[IntegerDigits[n]]]];
rev[n_]:=Reverse[IntegerDigits[n]];
updown[n_]:=FromDigits[rev[n]];
mirror[n_]:=FromDigits[rev[n]/.{2-> 5,5-> 2}];
updownmirror[n_]:=FromDigits[rev[mirror[n]]];
lst2=Select[Range@188801,And[startQ[#],subQ[#],PrimeQ[#],PrimeQ[updown[#]],PrimeQ[mirror[#]],PrimeQ[updownmirror[#]]]&];
Join[lst1,lst2] (* Ivan N. Ianakiev, Oct 08 2015 *)

Python

from sympy import isprime
from itertools import count, islice, product
def t(s): return s.translate({ord("2"):ord("5"), ord("5"):ord("2")})
def ok(s): # s is a string of digits
    return all(isprime(int(w)) for w in [s, s[::-1], t(s), t(s[::-1])])
def agen(): # generator of terms
    yield from (2, 5)
    for d in count(2):
        for mid in product("01258", repeat=d-2):
            s = "1" + "".join(mid) + "1"
            if ok(s): yield int(s)
print(list(islice(agen(), 35))) # Michael S. Branicky, Apr 27 2024

Extensions

5 added by Patrick Capelle, Feb 06 2008