%I #17 Jun 30 2023 16:32:23
%S 347,743,15581,42451,51581,54421,58151,58511,81551,112583,115823,
%T 118253,121853,122443,123581,125183,125813,128153,128351,132851,
%U 135281,138251,144223,152183,152381,153281,158231,181253,181523,185123,211583,214243,215183,215381,218513,218531,223441,235181,235811,238151,242413
%N Prime numbers such that the product of their digits equals twice the number of their digits times the sum of their digits.
%H Michael S. Branicky, <a href="/A343701/b343701.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%p q:= n-> (l-> mul(i,i=l)=2*nops(l)*add(i,i=l))(convert(n, base, 10)):
%p select(q, [ithprime(j)$j=1..100000])[]; # _Alois P. Heinz_, May 30 2021
%t Select[Range[1000000], PrimeQ[#] && Times@@IntegerDigits[#] == 2 Length[IntegerDigits[#]] Total[IntegerDigits[#]] &]
%t Select[Prime[Range[22000]],Times@@IntegerDigits[#]==2(IntegerLength[#]Total[ IntegerDigits[ #]])&] (* _Harvey P. Dale_, Jun 30 2023 *)
%o (Python)
%o from math import prod
%o from sympy import isprime
%o from sympy.utilities.iterables import multiset_permutations as mp
%o from itertools import count, islice, combinations_with_replacement as mc
%o def c(s):
%o d = list(map(int, s))
%o return prod(d) == 2*len(d)*sum(d)
%o def agen():
%o for d in count(2):
%o okset = set()
%o for cand in ("".join(m) for m in mc("987654321", d)):
%o if c(cand):
%o for p in mp(cand, d):
%o t = int("".join(p))
%o if isprime(t): okset.add(t)
%o yield from sorted(okset)
%o print(list(islice(agen(), 41))) # _Michael S. Branicky_, Nov 30 2022
%Y Cf. A064155.
%K nonn,base
%O 1,1
%A _Tanya Khovanova_, May 26 2021