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A346275 Number of n-digit primes with digital product = 3.

%I A346275 #27 Jul 20 2021 04:02:01
%S A346275 1,2,3,0,3,3,0,4,4,0,5,2,0,1,0,0,1,2,0,3,1,0,2,1,0,0,1,0,4,3,0,0,1,0,
%T A346275 4,3,0,1,6,0,0,4,0,2,1,0,2,6,0,2,2,0,5,3,0,4,3,0,3,1,0,1,3,0,1,4,0,0,
%U A346275 2,0,3,4,0,4,1,0,2,4,0,3,2,0,1,6,0,2,3,0,3,4,0
%N A346275 Number of n-digit primes with digital product = 3.
%C A346275 Equivalently: Number of n-digit terms of A107689 that are primes.
%C A346275 This sequence is similar to A346274 where digital product = 7.
%C A346275 a(n) = n if n = 1 or n = 2 or n = 3.
%C A346275 Conjecture: for n >= 4, there is at least one composite number among the n-digit numbers with digital product = 3 <==> a(n) < n for n >= 4 (guess proposed by Derek Holton in reference).
%C A346275 Also a(n) = 0 for n = 15, 26, 32, 68, ...
%D A346275 Derek Holton, A Second Step to Mathematical Olympiad Problems, Vol. 7, Mathematical Olympiad Series, World Scientific, 2011, Section 8.14 - 69, page 287.
%F A346275 a(3*k+1) = 0 for k > 0.
%e A346275 3 is prime, hence a(1) = 1.
%e A346275 13 and 31 are primes, hence a(2) = 2.
%e A346275 11113, 11131, 11311 are primes, but 13111 =  7*1873 and 31111 = 53*587, hence a(5) = 3.
%t A346275 a[n_] := Count[(10^n - 1)/9 + 2*10^Range[0, n - 1], _?PrimeQ]; Array[a, 100] (* _Amiram Eldar_, Jul 13 2021 *)
%o A346275 (Python)
%o A346275 from sympy import isprime
%o A346275 def a(n): return sum(isprime(int('1'*(n-1-i)+'3'+'1'*i)) for i in range(n))
%o A346275 print([a(n) for n in range(1, 92)]) # _Michael S. Branicky_, Jul 13 2021
%Y A346275 Cf A107689, A346274.
%K A346275 nonn,base
%O A346275 1,2
%A A346275 _Bernard Schott_, Jul 13 2021