A245017 Numbers k such that (product of digits of k) + 1 and (product of digits of k)^2 + 1 are both prime.

1, 2, 4, 6, 11, 12, 14, 16, 21, 22, 23, 25, 28, 32, 41, 44, 49, 52, 58, 61, 66, 82, 85, 94, 111, 112, 114, 116, 121, 122, 123, 125, 128, 132, 141, 144, 149, 152, 158, 161, 166, 182, 185, 194, 211, 212, 213, 215, 218, 221, 224, 229, 231, 236, 242, 245, 251, 254, 263, 279, 281, 292

Offset: 1

Author

Derek Orr, Jul 12 2014

Keywords

• nonn
• base

Comments

A number k is a term of this sequence iff A007954(k) and A007954(k)^2 are both in A006093.

This sequence is infinite. With any number a(n), you can add infinitely many 1's to its decimal representation. E.g., 82 is in this sequence, so 821, 812, 1182, 18112, 81211, etc. are also terms of this sequence.

Examples

			(9*4) + 1 = 37 is prime and (9*4)^2 + 1 = 1297 is prime. Thus 94 is a term of this sequence.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007954, A009994, A081988, A244748.

Programs

Mathematica

bpQ[n_]:=Module[{c=Times@@IntegerDigits[n]},AllTrue[{c+1,c^2+1},PrimeQ]]; Select[Range[300],bpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 09 2019 *)

PARI

for(n=1, 10^3, d=digits(n); p=prod(i=1, #d, d[i]); if(ispseudoprime(p+1) && ispseudoprime(p^2 + 1), print1(n,", ")))