A370850 Lesser of two consecutive primes whose digits' products are also prime.

2, 3, 5, 13, 11113, 111111113, 11111111111111111111111111117

Offset: 1

Author

Claude H. R. Dequatre, Mar 03 2024

Keywords

• nonn
• base
• hard

Comments

If it exists, a(8) > 10^500. - Hugo Pfoertner, Mar 03 2024

If it exists, a(8) > 10^10000. - Bert Dobbelaere, Mar 16 2024

Examples

			13 is a term because 13 is prime, the product of its digits is 3 which is prime and the product of the digits of 17, the next prime to 13, is 7 and 7 is prime.
19 is not a term because the product of its digits is 9 and 9 is not prime.
131 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 137, the next prime to 131, is 21 and 21 is not prime.
		

Crossrefs

Cf. A000040, A053666.

Cf. also A370848, A370851.

Programs

Mathematica

Select[Prime[Range[10^5]], PrimeQ[Apply[Times, IntegerDigits[#]]]&&PrimeQ[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) && isprime(y);
forprime(p=2,20000,if(isok(p),print1(p", ")))

PARI

a370850(maxdigits=100) = {my(L=List()); for (n=1, maxdigits, my (r=(10^n-1)/9, d=digits(r)); foreach ([2,3,5,7], s, for (k=1, #d, my(dd=d); dd[k]=s; my (q=fromdigits(dd)); if (ispseudoprime(q) && isprime(vecprod(digits(nextprime(q+1)))), listput(L,q))))); vecsort(Vec(L))};
a370850() \\ Hugo Pfoertner, Mar 03 2024

Extensions

a(7) from Hugo Pfoertner, Mar 03 2024