A390697 Primes k for which the concatenation, in ascending order, of all prime digit permutations, yields a prime.

2, 3, 5, 7, 11, 19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 127, 151, 191, 211, 223, 227, 229, 233, 241, 257, 263, 269, 271, 313, 331, 353, 367, 383, 409, 421, 431, 433, 443, 449, 457, 463, 487, 499, 503, 509, 523, 541, 547, 557, 569, 577, 599, 607, 643, 659

Offset

1,1

Comments

If a prime p belongs to the sequence, then every prime digit permutation of p (excluding some primes with a leading zero) also belongs to the sequence.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..4137 (terms 1..1049 from Jean-Marc Rebert)

Example

127 is a term because the concatenation, in ascending order, of all prime permutations of its digits (namely, 127 and 271) yields 127271, which is also prime.
1019 is a term because the concatenation, in ascending order, of all prime permutations of its digits (namely, 191, 911, 1019, 1091, 1109, 1901 and 9011) yields 19191110191091110919019011, which is also prime.
13 is not a term since 1331 = 11^3 is not prime.

Mathematica

Select[Prime[Range[120]], PrimeQ[FromDigits[IntegerDigits[Sort[Select[FromDigits/@Permutations[IntegerDigits[#]], PrimeQ]]]//Flatten]]&] (* James C. McMahon, Dec 02 2025 *)

Prog

(Python)
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def ok(n): return isprime(n) and isprime(int("".join(str(t) for p in multiset_permutations(str(n)) if isprime(t:=int("".join(p))))))
print([k for k in range(700) if ok(k)]) # Michael S. Branicky, Nov 15 2025

Crossrefs

Cf. A000040, A003459, A039999, A046811, A072857, A262988.

Sequence in context: A262837 A143260 A246044 * A391021 A039986 A278694

Adjacent sequences: A390694 A390695 A390696 * A390698 A390699 A390700

Keyword

nonn,base

Author

Jean-Marc Rebert, Nov 15 2025

Status

approved