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

1,1

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?

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Intersection of A006567 and A080790.

Subset of A136634.

Cf. A004086, A030101, A081434.

Sequence in context: A309720 A233435 A049742 * A247867 A113601 A158864

Adjacent sequences: A347206 A347207 A347208 * A347210 A347211 A347212

Keyword

nonn,base

Author

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

Status

approved