A262729 Strong (2,3,5,7)-primes. (See Comments for precise definition.)

2, 171472673, 343808687, 1364225981, 1469999801, 1871684753, 2110769237, 2227044401, 2411201729, 2485782361, 2545607453, 3795488227, 3946237717, 4213334953, 4395443513, 5308651577, 5770033901, 5832097819, 6385775491, 6694883219, 7064806421, 7235208829

Offset: 1

Author

Clark Kimberling, Oct 03 2015

Keywords

• nonn
• base

Comments

Let V = (b(1), b(2), ..., b(k)), where k > 1 and b(i) are distinct integers > 1 for j = 1..k. Call p a V-prime if the digits of p in base b(1) spell a prime in each of the bases b(2), ..., b(k). Call p a strong V-prime if p is a (b(j), ..., b(k))-prime for each of the tuples (b(j), ..., b(k)), for j = 1..k-1.

a(157) > 10^11. - Hiroaki Yamanouchi, Oct 25 2015

Examples

			Let p = 171472673. Confirmation that p is a strong (2,3,5,7)-prime follows.
Base-2 for p: u = (1,0,1,0,0,0,1,1,1,0,0,0,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,1);
u in base 3 spells the prime 8488002487771;
u in base 5 spells the prime 7749195106457425001;
u is base 7 spells the prime 67054080721013093290423.
Base-3 for p: v = (1, 0, 2, 2, 2, 1, 1, 2, 2, 2, 0, 1, 0, 2, 1, 2, 0, 2);
v in base 5 spells the prime 838940251427;
v in base 7 spells the prime 243692337097757.
Base-5 for p: w = (3, 2, 2, 3, 4, 4, 1, 1, 1, 1, 4, 3);
w in base 7 spells the prime 6598716743.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..156

Crossrefs

Cf. A000040, A262727, A262728.

Programs

Mathematica

{b1, b2, b3, b4} = {2, 3, 5, 7}; z = 10000000;
Select[Prime[Range[z]],
PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b4]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b4]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b4]] &]
(* Peter J. C. Moses, Sep 27 2015 *)

Extensions

a(4)-a(22) from Hiroaki Yamanouchi, Oct 25 2015