A359137 -id:A359137 - cp's OEIS frontend

A359139 A359137 together with 2, 3, 5, 7.

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 107, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 419, 421

Offset: 1

N. J. A. Sloane and Allan C. Wechsler, Jan 22 2023

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A359136, A359137, A359138.

More than the usual number of terms are shown in order to distinguish this from other similar sequences.

Incorrect terms removed by Andrew Howroyd, Jan 22 2023

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.

Original entry on oeis.org

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

Rémy Sigrist, Jan 23 2023

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.

			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.

Cf. A359136, A359137, A360040.

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)) }

A359136 Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 1).

Original entry on oeis.org

11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 419, 421

Offset: 1

N. J. A. Sloane and Allan C. Wechsler, Jan 22 2023

A prime p with decimal expansion p = d_1 d_2 ... d_m is in this sequence iff there is a non-identity permutation pi in S_m such that q = d_pi(1) d_pi(2) ... d_pi(m) is also a prime. The prime q may or may not be equal to p.  Leading zeros are permitted in q.
One must be careful when using the phrase "nontrivial permutation of the digits". When the first and third digits of 101 are exchanged, this is applying the nontrivial permutation (1,3) in S_3 to the digits, leaving the number itself unchanged. One should specify whether it is the permutation that is nontrivial, or its action on what is being permuted. In this sequence and A359137, we mean that the permutation itself is nontrivial.
There are only 34 primes not in this sequence, the greatest of which is 5849. - Andrew Howroyd, Jan 22 2023

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Many OEIS entries are subsequences (possibly after omitting 2, 3, 5, and 7): A007500, A055387, A061461, A069706, A090933, A225035.

Cf. A359137, A359138, A359139.

isok(n)={my(v=vecsort(digits(n))); if(#Set(v)<#v, 1, forperm(v, u, my(t=fromdigits(Vec(u))); if(isprime(t) && t<>n, return(1))); 0)} \\ Andrew Howroyd, Jan 22 2023

from sympy import isprime
from itertools import permutations as P
def ok(n):
    if not isprime(n): return False
    if len(s:=str(n)) > len(set(s)): return True
    return any(isprime(t) for t in (int("".join(p)) for p in P(s)) if t!=n)
print([k for k in range(422) if ok(k)]) # Michael S. Branicky, Jan 23 2023

More than the usual number of terms are shown in order to distinguish this from neighboring sequences.

Incorrect terms removed by Andrew Howroyd, Jan 22 2023

A359138 A359136 together with 2, 3, 5, 7.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 419, 421

Offset: 1

N. J. A. Sloane and Allan C. Wechsler, Jan 22 2023

By including the "by convention" terms 2, 3, 5, and 7, many sequences such as A007500 are now subsequences.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A359136, A359137, A359139.

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.

Original entry on oeis.org

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

Rémy Sigrist, Jan 23 2023

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.

Cf. A359136, A359137, A360041.

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)) }

The nontrivial permutations of the digits of 409 (permitting leading zeros) are:
= 7^2,
= 2 * 47,
= 2 * 5 * 7^2,
= 2^3 * 113,
= 2^2 * 5 * 47,
so 409 belongs to the sequence.

cp's OEIS Frontend

A359139 A359137 together with 2, 3, 5, 7.

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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.

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A359136 Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 1).

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A359138 A359136 together with 2, 3, 5, 7.

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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.

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