A383907 Echo primes: primes p such that the greatest prime factor of p-1 is a suffix of p.

13, 73, 127, 163, 193, 197, 313, 337, 419, 433, 757, 929, 1153, 2017, 2311, 2593, 2647, 3137, 3659, 4483, 4673, 5741, 6857, 7057, 12071, 12097, 13267, 13313, 13619, 14407, 15877, 17191, 18041, 18433, 18439, 19273, 19531, 20353, 21319, 21961, 22279, 24103, 24697, 25411

Offset: 1

Author

Giorgos Kalogeropoulos, May 15 2025

Keywords

• nonn
• base

Comments

These are primes in A383896(n).

The first pair of twin echo primes is 6467806769 and 6467806771.

Conjecture: There are infinitely many echo primes.

Crossrefs

Cf. A000040, A006530, A383896.

Programs

Mathematica

Select[Prime@Range@3000,(f=FactorInteger[#-1][[-1,1]]; Mod[#,10^IntegerLength@f]==f)&]

Python

from sympy import factorint, isprime
def ok(n): return n > 2 and isprime(n) and str(n).endswith(str(max(factorint(n-1))))
print([k for k in range(30000) if ok(k)]) # Michael S. Branicky, May 15 2025