A360040 Prime numbers missing from A359136: prime numbers for which none of the nontrivial permutations of its digits (permitting leading zeros) produces a prime number.

2, 3, 5, 7, 19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 257, 263, 269, 409, 431, 487, 523, 541, 827, 829, 853, 859, 2861, 4027, 4801, 5209, 5623, 5849

Offset: 1

Author

Rémy Sigrist, Jan 23 2023

Keywords

• nonn
• base
• fini
• full

Comments

Any prime number p >= 10^11 has necessarily a duplicate digit, say that appears at positions i and j. Applying the nontrivial permutation (i j) to the digits of p yields a prime number (p itself), hence p does not belong to the sequence and the sequence is finite.

All terms belong to A360041.

Crossrefs

Cf. A359136, A359137, A360041.

Programs

PARI

is(p) = { my (d=digits(p)); if (#d > #Set(d), return (0), forperm (vecsort(d), t, my (q=fromdigits(Vec(t))); if (p!=q && isprime(q), return (0))); return (1)) }

Formula

The nontrivial permutations of the digits of 409 (permitting leading zeros) are:

049 = 7^2,

094 = 2 * 47,

490 = 2 * 5 * 7^2,

904 = 2^3 * 113,

940 = 2^2 * 5 * 47,

so 409 belongs to the sequence.