A133187 Prime numbers formed by the concatenation of q and p, where q > p are also primes.

53, 73, 113, 137, 173, 193, 197, 233, 293, 313, 317, 373, 433, 593, 613, 617, 673, 677, 733, 797, 977, 1013, 1033, 1093, 1097, 1277, 1373, 1493, 1637, 1733, 1913, 1933, 1973, 1993, 1997, 2113, 2237, 2273, 2293, 2297, 2311, 2333, 2393, 2417, 2633, 2693, 2713

Offset: 1

Author

Tom Mueller (muel4503(AT)uni-trier.de), Dec 17 2007

Keywords

• base
• nonn

Comments

These numbers are called Caesar primes because the birth date of Julius Caesar (July 13th) provides one example of such a number, i.e. p=7 and q=13 give the prime 137.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A019549, A105184.

Programs

Mathematica

lim=2700;plim=Max[FromDigits[Rest[IntegerDigits[lim]]],FromDigits[Drop[IntegerDigits[lim],-1]]];f2p[{p_,q_}]:=FromDigits[Join[IntegerDigits[q],IntegerDigits[p]]];p=Prime[Range[PrimePi[plim]]];p2=Subsets[p,{2}];Union[Select[f2p/@p2,PrimeQ[#]&&#<=lim&]] (* James C. McMahon, Mar 12 2025 *)

Python

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return any(s[i]!='0' and (q:=int(s[:i])) > (p:=int(s[i:])) and isprime(q) and isprime(p) for i in range(1, len(s)))
print([k for k in range(2800) if ok(k)]) # Michael S. Branicky, Apr 05 2025

Extensions

a(27)-a(47) from James C. McMahon, Mar 12 2025