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

17, 31, 71, 113, 131, 151, 211, 311, 1117, 1151, 1171, 1511, 2111, 11117, 11131, 11171, 11311, 111121, 111211, 112111, 113111, 131111, 311111, 511111, 1111151, 1111211, 1111711, 1117111, 1171111, 11111117, 11111131, 11111171, 11111311, 11113111, 11131111, 71111111

Offset: 1

Author

Claude H. R. Dequatre, Mar 03 2024

Keywords

• nonn
• base

Examples

			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.
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.
29 is not a term because the product of its digits is 18 and 18 is not prime.
		

Crossrefs

Cf. A000040, A002808, A053666.

Cf. also A370848, A370850.

Programs

Mathematica

Select[Prime[Range[6*10^6]], PrimeQ[Apply[Times, IntegerDigits[#]]]&&CompositeQ[Apply[Times,IntegerDigits[NextPrime[#]]]]&] (* James C. McMahon, Mar 03 2024 *)

PARI

isok(p)=my(x=vecprod(digits(p)),y=vecprod(digits(nextprime(p+1))));isprime(x) && y>3 &&!isprime(y);
forprime(p=2,20000,if(isok(p),print1(p", ")))

Python

from math import prod
from itertools import count, islice
from sympy import isprime, nextprime
def A370851_gen(): # generator of terms
    for l in count(1):
        k = (10**l-1)//9
        for m in range(l):
            a = 10**m
            for j in (1,2,4,6):
                p = k+a*j
                if isprime(p) and not (isprime(s:=prod(map(int,str(nextprime(p))))) or s==1):
                    yield p
A370851_list = list(islice(A370851_gen(),20)) # Chai Wah Wu, Mar 25 2024