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

1,1

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

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 *)

Prog

(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

Crossrefs

Cf. A019549, A105184.

Sequence in context: A045807 A007644 A136073 * A057667 A160041 A013538

Adjacent sequences: A133184 A133185 A133186 * A133188 A133189 A133190

Keyword

base,nonn

Author

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

Extensions

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

Status

approved