A375965 Primes which can be turned into a different prime by exchanging two consecutive digits.

13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 239, 241, 251, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 349, 373, 379, 389, 397, 401, 419, 421, 439, 457, 461, 463, 467, 491, 503

Offset: 1

Author

Matthew House, Sep 04 2024

Keywords

• nonn
• base
• easy

Comments

Heuristically, both this sequence and its complement should have positive density within the primes. Empirically, n/pi(a(n)) approaches ~0.75 for large n. But is this sequence even infinite?

Dickson's conjecture implies it is infinite, e.g. it implies that there are infinitely many primes ending in 13 for which changing the last two digits to 31 produces a prime. - Robert Israel, Sep 04 2024

Examples

			2, 3, 5, and 7 are excluded, since they have no two digits to exchange.
11 is excluded, since it yields only itself.
13 and 17 are included, since they yield the primes 31 and 71.
19, 23, and 29 are excluded, since they yield the composite numbers 91, 32, and 92.
		

Links

• Matthew House, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A225035.

Programs

Maple

filter:= proc(n) local L,d,i;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10); d:= nops(L);
  for i from 1 to d-1 do
      if L[i] <> L[i+1] and isprime(n + (L[i]-L[i+1])*(10^i-10^(i-1))) then return true fi
  od;
  false
end proc:
select(filter, [seq(i,i=11 .. 1---,2)]); # Robert Israel, Sep 04 2024

Mathematica

Select[Prime[Range[100]], With[{digits = IntegerDigits[#]}, AnyTrue[Complement[FromDigits[Permute[digits, Cycles[{{#, # + 1}}]]] & /@ Range[Length[digits] - 1], {#}], PrimeQ]] &]

Python

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    for i in range(len(s)-1):
        t = int(s[:i]+s[i+1]+s[i]+s[i+2:])
        if t != n and isprime(t): return True
    return False
print([k for k in range(504) if ok(k)]) # Michael S. Branicky, Sep 04 2024