A360041 Prime numbers missing from A359137: prime numbers for which none of the nontrivial permutations of its digits (not permitting leading zeros) produces a prime number.

2, 3, 5, 7, 19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 103, 109, 257, 263, 269, 307, 401, 409, 431, 487, 503, 509, 523, 541, 601, 607, 809, 827, 829, 853, 859, 2017, 2087, 2861, 4027, 4051, 4079, 4801, 5021, 5209, 5623, 5849, 6047, 6053, 6803, 8053, 8059

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.

Examples

			The nontrivial permutations of the digits of 607 (not permitting leading zeros) are:
  670 = 2 * 5 * 67,
  706 = 2 * 353,
  760 = 2^3 * 5 * 19,
so 607 belongs to the sequence.
		

Crossrefs

Cf. A359136, A359137, A360040.

Programs

PARI

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