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

Author

James S. DeArmon, Sep 15 2024

Keywords

• nonn
• base

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".)

Examples

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

Crossrefs

Subsequence of A225035. Cf. A375965.

Programs

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

Extensions

Terms corrected by Michael S. Branicky, Sep 17 2024