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A370851 Lesser of two consecutive primes such that the product of its digits is also prime and that of the other is composite.

%I A370851 #18 Mar 26 2024 01:59:27
%S A370851 17,31,71,113,131,151,211,311,1117,1151,1171,1511,2111,11117,11131,
%T A370851 11171,11311,111121,111211,112111,113111,131111,311111,511111,1111151,
%U A370851 1111211,1111711,1117111,1171111,11111117,11111131,11111171,11111311,11113111,11131111,71111111
%N A370851 Lesser of two consecutive primes such that the product of its digits is also prime and that of the other is composite.
%e A370851 17 is a term because 17 is prime, the product of its digits is 7 which is prime and the product of the digits of 19, the next prime to 17, is 9 and 9 is composite.
%e A370851 13 is not a term because although it is prime and the product of its digits is 3 which is also prime, the product of the digits of 17, the next prime to 13, is 7 and 7 is not composite.
%e A370851 29 is not a term because the product of its digits is 18 and 18 is not prime.
%t A370851 Select[Prime[Range[6*10^6]], PrimeQ[Apply[Times, IntegerDigits[#]]]&&CompositeQ[Apply[Times,IntegerDigits[NextPrime[#]]]]&] (* _James C. McMahon_, Mar 03 2024 *)
%o A370851 (PARI) isok(p)=my(x=vecprod(digits(p)),y=vecprod(digits(nextprime(p+1))));isprime(x) && y>3 &&!isprime(y);
%o A370851 forprime(p=2,20000,if(isok(p),print1(p", ")))
%o A370851 (Python)
%o A370851 from math import prod
%o A370851 from itertools import count, islice
%o A370851 from sympy import isprime, nextprime
%o A370851 def A370851_gen(): # generator of terms
%o A370851     for l in count(1):
%o A370851         k = (10**l-1)//9
%o A370851         for m in range(l):
%o A370851             a = 10**m
%o A370851             for j in (1,2,4,6):
%o A370851                 p = k+a*j
%o A370851                 if isprime(p) and not (isprime(s:=prod(map(int,str(nextprime(p))))) or s==1):
%o A370851                     yield p
%o A370851 A370851_list = list(islice(A370851_gen(),20)) # _Chai Wah Wu_, Mar 25 2024
%Y A370851 Cf. A000040, A002808, A053666.
%Y A370851 Cf. also A370848, A370850.
%K A370851 nonn,base
%O A370851 1,1
%A A370851 _Claude H. R. Dequatre_, Mar 03 2024