A361530 Primes that can be written as the result of shuffling the decimal digits of two primes.
23, 37, 53, 73, 113, 127, 131, 137, 139, 151, 157, 173, 179, 193, 197, 211, 223, 229, 233, 239, 241, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 421, 431, 433, 457, 523, 541, 547, 571, 593, 613, 617, 631, 673, 677, 719
Offset: 1
Examples
37 and 73 are in the sequence because they are both the result of shuffling 3 and 7. 127 is in the sequence because it is the result of shuffling 2 and the digits of 17. 1193 is in the sequence because it is the result of shuffling the digits of 13 and the digits of 19. 163 is not in the sequence because it is not the result of shuffling the digits of two primes. 163 is the result of permuting the digits of 3 and 61; however, 163 contains the digits of 61 in the wrong order.
Links
- Michael S. Branicky, Table of n, a(n) for n = 1..10000
- John D. Cook, Shuffle product.
- Wikipedia, Shuffle product.
Programs
-
Python
import sympy def get_shuffle_product(list_1, list_2): shuffle_product = set() shuffle = [] _get_shuffle_product(list_1, list_2, shuffle, shuffle_product) return shuffle_product def _get_shuffle_product(list_1, list_2, shuffle, shuffle_product): if len(list_1) == 0 and len(list_2) == 0: shuffle_product.add(tuple(shuffle)) return else: if len(list_1) == 0: shuffle.append(list_2[0]) _get_shuffle_product(list_1, list_2[1:], shuffle, shuffle_product) shuffle.pop() elif len(list_2) == 0: shuffle.append(list_1[0]) _get_shuffle_product(list_1[1:], list_2, shuffle, shuffle_product) shuffle.pop() else: shuffle.append(list_1[0]) _get_shuffle_product(list_1[1:], list_2, shuffle, shuffle_product) shuffle.pop() shuffle.append(list_2[0]) _get_shuffle_product(list_1, list_2[1:], shuffle, shuffle_product) shuffle.pop() max_prime_index = 25 # one and two digit primes. max_element = 999 prime_set = set() for p_index in range(1, max_prime_index+1): p = sympy.prime(p_index) for q_index in range(p_index, max_prime_index+1): q = sympy.prime(q_index) list_p = list(str(p)) list_q = list(str(q)) shuffle_product = get_shuffle_product(list_p, list_q) for s in shuffle_product: candidate = int(''.join(s)) if sympy.isprime(candidate) and candidate <= max_element: prime_set.add(candidate) print(sorted(prime_set)) -
Python
from sympy import isprime from itertools import chain, combinations def powerset(s): # skipping empty set and entire set return chain.from_iterable(combinations(s, r) for r in range(1, len(s))) def ok(n): if not isprime(n): return False s = str(n) for indices in powerset(range(len(s))): t1 = "".join(s[i] for i in indices) t2 = "".join(s[i] for i in range(len(s)) if i not in indices) if t1[0] != "0" and t2[0] != "0" and isprime(int(t1)) and isprime(int(t2)): return True print([k for k in range(720) if ok(k)]) # Michael S. Branicky, Apr 16 2023
Comments