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

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