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

1,1

Comments

Each term is essentially an element of the shuffle product of the decimal digits of two primes (possibly equal).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

John D. Cook, Shuffle product.

Wikipedia, Shuffle product.

Example

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.

Prog

(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

Crossrefs

Cf. A019549, A083427, A105184.

Sequence in context: A329262 A268343 A063643 * A057876 A244282 A051362

Adjacent sequences: A361527 A361528 A361529 * A361531 A361532 A361533

Keyword

nonn,base

Author

Robert C. Lyons, Mar 14 2023

Status

approved