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 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. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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). LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 EXAMPLE 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). 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 *) PROG (Python 3) 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 CROSSREFS Cf. A009994. Subsequence of A038618. Sequence in context: A082807 A088288 A142068 * A141920 A124308 A059702 Adjacent sequences: A350571 A350572 A350573 * A350575 A350576 A350577 KEYWORD nonn,base AUTHOR William Riley Barker, Jan 06 2022 STATUS approved

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Last modified February 26 17:19 EST 2024. Contains 370352 sequences. (Running on oeis4.)