A343701 Prime numbers such that the product of their digits equals twice the number of their digits times the sum of their digits.

347, 743, 15581, 42451, 51581, 54421, 58151, 58511, 81551, 112583, 115823, 118253, 121853, 122443, 123581, 125183, 125813, 128153, 128351, 132851, 135281, 138251, 144223, 152183, 152381, 153281, 158231, 181253, 181523, 185123, 211583, 214243, 215183, 215381, 218513, 218531, 223441, 235181, 235811, 238151, 242413

Offset: 1

Author

Tanya Khovanova, May 26 2021

Keywords

• nonn
• base

Examples

			347 is a 3-digit prime number. The product of its digits is 84. The sum of its digits is 14. As 84 = 2*3*14, this number is in the sequence.
		

Links

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

Crossrefs

Cf. A064155.

Programs

Maple

q:= n-> (l-> mul(i,i=l)=2*nops(l)*add(i,i=l))(convert(n, base, 10)):
select(q, [ithprime(j)$j=1..100000])[];  # Alois P. Heinz, May 30 2021

Mathematica

Select[Range[1000000], PrimeQ[#] && Times@@IntegerDigits[#] == 2 Length[IntegerDigits[#]] Total[IntegerDigits[#]] &]
Select[Prime[Range[22000]],Times@@IntegerDigits[#]==2(IntegerLength[#]Total[ IntegerDigits[ #]])&] (* Harvey P. Dale, Jun 30 2023 *)

Python

from math import prod
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def c(s):
    d = list(map(int, s))
    return prod(d) == 2*len(d)*sum(d)
def agen():
    for d in count(2):
        okset = set()
        for cand in ("".join(m) for m in mc("987654321", d)):
            if c(cand):
                for p in mp(cand, d):
                    t = int("".join(p))
                    if isprime(t): okset.add(t)
        yield from sorted(okset)
print(list(islice(agen(), 41))) # Michael S. Branicky, Nov 30 2022