A350574 Primes p such that p, asc(p), and desc(p) are all distinct primes (where asc(p) and desc(p) are the digits of p in ascending and descending order, respectively) and p is the minimum of all permutations of its digits with this property.

131, 197, 373, 419, 571, 593, 617, 839, 919, 1297, 1327, 1429, 1879, 1949, 1993, 2129, 2213, 2591, 3539, 4337, 4637, 4639, 5519, 6619, 8389, 8933, 11491, 11519, 11527, 11597, 11897, 11969, 12757, 12829, 12979, 13649, 13879, 14537, 14737, 14741, 14891

Offset: 1

Author

William Riley Barker, Jan 06 2022

Keywords

• nonn
• base

Comments

All terms in this sequence are zero-free: If p contained 0 as a digit, then desc(p) would end with the zero digit, making it divisible by 10 (thus composite).

Examples

			131 is prime, and so are asc(131) = 113 and desc(131) = 311. Further, these are all distinct primes.
419, asc(419) = 149, and desc(419) = 941 are all distinct primes, and 419 is the smallest permutation of the digits {1,4,9} with this property (149 is not included because asc(149) = 149, so these would not be distinct; 491 is not included because 419 < 491 with the same set of digits).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A009994.

Subsequence of A038618.

Programs

Maple

d:= n-> convert(n, base, 10):
g:= (n, r)-> parse(cat(sort(d(n), r)[])):
f:= n-> (s-> nops(s)=3 and andmap(isprime, s))({n, g(n, `<`), g(n, `>`)}):
q:= n-> f(n) and n=min(select(f, map(x-> parse(cat(x[])),
        combinat[permute](d(n))))):
select(q, [$1..15000])[];  # Alois P. Heinz, Jan 15 2022

Mathematica

Select[Prime@Range@2000,AllTrue[FromDigits/@{s=Sort[d=IntegerDigits@#],Reverse@s},PrimeQ]&&Min@Most@Rest@Sort@Select[FromDigits/@Permutations[d],PrimeQ]==#&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

Python

import numpy as np
# preliminary functions we will use in building our list
def is_prime(n):
    for d in range(2,int(np.sqrt(n))+1):
        if n % d == 0:
            return False
    return True
def asc(n): # returns integer with digits of n in ascending order
    n_list = [int(digit) for digit in str(n)] # separate digits of n into a list
    n_list.sort() # rearrange numbers in ascending order
    asc_n = int(''.join([str(digit) for digit in n_list])) # concatenate the sorted digits
    return asc_n
def desc(n): # returns integer with digits of n in descending order
    n_list = [int(digit) for digit in str(n)]
    n_list.sort(reverse=True)
    desc_n = int(''.join([str(digit) for digit in n_list]))
    return desc_n
N = 5
# get list of integers n such that n, asc(n), and desc(n) are all distinct primes
condition_1 = [n for n in range(2,10**N)
               if is_prime(n)
               and is_prime(asc(n))
               and is_prime(desc(n))
               and n not in (asc(n),desc(n))]
# refine so that our list includes only the minimum permutation of a given set of digits satisfying condition 1
condition_2 = []
for num in condition_1:
    if asc(num) not in [asc(n) for n in condition_2]:
        condition_2.append(num)
print(condition_2)

Python

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A350574_gen(): # generator of terms
    for l in count(1):
        rlist = []
        for a in combinations_with_replacement('123456789',l):
            s = ''.join(a)
            p, q = int(s), int(s[::-1])
            if p != q and isprime(p) and isprime(q):
                for b in multiset_permutations(a):
                    r = int(''.join(b))
                    if p < r < q and isprime(r):
                        rlist.append(r)
                        break
        yield from sorted(rlist)
A350574_list = list(islice(A350574_gen(),50)) # Chai Wah Wu, Feb 13 2022