A376213 Prime numbers wherein a triple exchange of 3 of the digits creates two prime numbers, neither of which has a leading zero digit.

113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1013, 1019, 1031, 1091, 1123, 1163, 1181, 1193, 1213, 1231, 1237, 1279, 1297, 1307, 1319, 1321, 1327, 1399, 1439, 1487, 1499, 1543, 1549, 1571, 1613, 1621, 1637, 1733, 1747, 1759, 1777, 1811, 1831, 1913

Offset

1,1

Comments

A triple exchange is a permutation of 3 elements in which all 3 three items change position. (For the triple "ABC", that would be "BCA" and "CAB".)

The three prime numbers must be distinct, e.g. we don't allow permuting the three 1's of 1117. - Robert Israel, Sep 15 2025

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The first term is the prime 113, since 131 and 311 are also prime. Another term is 1013, since 1103 and 1301 are prime.

Maple

filter:= proc(n) local L, d, p, n2, n3;
  if not isprime(n) then return false fi;
  L:= convert(n, base, 10);
  d:= nops(L);
  for p in combinat:-choose(d, 3) do
    n2:= n + 10^(p[1]-1)*(L[p[2]]-L[p[1]]) + 10^(p[2]-1)*(L[p[3]]-L[p[2]]) + 10^(p[3]-1)*(L[p[1]]-L[p[3]]);
    n3:= n + 10^(p[1]-1)*(L[p[3]]-L[p[1]]) + 10^(p[2]-1)*(L[p[1]]-L[p[2]]) + 10^(p[3]-1)*(L[p[2]]-L[p[3]]);
    if n2 >= 10^(d-1) and n3 >= 10^(d-1) and nops({n, n2, n3})=3 and isprime(n2) and isprime(n3) then return true fi
  od;
  false
end proc:
select(filter, [seq(i, i=101 .. 2000, 2)]); # Robert Israel, Sep 15 2025

Prog

(Python)
from sympy import isprime
from itertools import combinations, permutations
def ok(n):
    if n < 100 or not isprime(n): return False
    s = list(str(n))
    for i, j, k in combinations(range(len(s)), 3):
        pset, w, x = {n}, s[:], s[:]
        w[i], w[j], w[k] = w[j], w[k], w[i]
        x[i], x[j], x[k] = x[k], x[i], x[j]
        if w[0] != "0" and isprime(t:=int("".join(w))): pset.add(t)
        if x[0] != "0" and isprime(t:=int("".join(x))): pset.add(t)
        if len(pset) == 3: return True
    return False
print([k for k in range(1920) if ok(k)]) # Michael S. Branicky, Sep 17 2024

Crossrefs

Subsequence of A225035. Cf. A375965.

Sequence in context: A180407 A163760 A179911 * A344627 A187867 A216288

Adjacent sequences: A376210 A376211 A376212 * A376214 A376215 A376216

Keyword

nonn,base

Author

James S. DeArmon, Sep 15 2024

Extensions

Terms corrected by Michael S. Branicky, Sep 17 2024

Status

approved