A347209 Emirps in both base 2 and base 10.

13, 37, 71, 97, 113, 167, 199, 337, 359, 701, 709, 739, 907, 937, 941, 953, 967, 1033, 1091, 1109, 1153, 1181, 1201, 1217, 1229, 1259, 1439, 1471, 1487, 1669, 1733, 1789, 1811, 1933, 1949, 3019, 3067, 3083, 3089, 3121, 3163, 3221, 3299, 3343, 3389, 3433, 3469, 3511, 3527, 3571, 3583, 3643, 3719

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 23 2021

Keywords

• nonn
• base

Comments

Primes p such that A030101(p) and A004086(p) are primes other than p.

Are there any cases where A030101(p) = A004086(p), i.e. emirps in A081434?

Examples

			a(3) = 71 is a term because 71 is prime, its base-10 reversal 17 is a prime other than 71, and its base-2 reversal 113 is a prime other than 71.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A006567 and A080790.

Subset of A136634.

Cf. A004086, A030101, A081434.

Programs

Maple

filter:= proc(n) local L,nL,i,r,s;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  nL:= nops(L);
  r:= add(10^(nL-i)*L[i],i=1..nL);
  if r=n or not isprime(r) then return false fi;
  L:= convert(n,base,2);
  nL:= nops(L);
  s:=add(2^(nL-i)*L[i],i=1..nL);
  s <> n and isprime(s)
end proc:
select(filter, [seq(i,i=3..10000,2)]);

Mathematica

Select[Range[4000], (ir = IntegerReverse[#]) != # && PrimeQ[#] && PrimeQ[ir] && (ir2 = IntegerReverse[#, 2]) != # && PrimeQ[ir2] &] (* Amiram Eldar, Aug 23 2021 *)
Select[Prime[Range[600]],!PalindromeQ[#]&&FromDigits[Reverse[IntegerDigits[#,2]],2]!=#&&AllTrue[{IntegerReverse[#],FromDigits[Reverse[IntegerDigits[#,2]],2]},PrimeQ]&] (* Harvey P. Dale, Oct 13 2022 *)

Python

from sympy import isprime, primerange
def ok(p):
    s, b = str(p), bin(p)[2:]
    if s == s[::-1] or b == b[::-1]: return False
    return isprime(int(s[::-1])) and isprime(int(b[::-1], 2))
print(list(filter(ok, primerange(1, 3720)))) # Michael S. Branicky, Aug 24 2021