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

%I A262729 #18 Nov 24 2015 01:42:01
%S A262729 2,171472673,343808687,1364225981,1469999801,1871684753,2110769237,
%T A262729 2227044401,2411201729,2485782361,2545607453,3795488227,3946237717,
%U A262729 4213334953,4395443513,5308651577,5770033901,5832097819,6385775491,6694883219,7064806421,7235208829
%N A262729 Strong (2,3,5,7)-primes. (See Comments for precise definition.)
%C A262729 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.
%C A262729 a(157) > 10^11. - _Hiroaki Yamanouchi_, Oct 25 2015
%H A262729 Hiroaki Yamanouchi, <a href="/A262729/b262729.txt">Table of n, a(n) for n = 1..156</a>
%e A262729 Let p = 171472673. Confirmation that p is a strong (2,3,5,7)-prime follows.
%e A262729 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);
%e A262729 u in base 3 spells the prime 8488002487771;
%e A262729 u in base 5 spells the prime 7749195106457425001;
%e A262729 u is base 7 spells the prime 67054080721013093290423.
%e A262729 Base-3 for p: v = (1, 0, 2, 2, 2, 1, 1, 2, 2, 2, 0, 1, 0, 2, 1, 2, 0, 2);
%e A262729 v in base 5 spells the prime 838940251427;
%e A262729 v in base 7 spells the prime 243692337097757.
%e A262729 Base-5 for p: w = (3, 2, 2, 3, 4, 4, 1, 1, 1, 1, 4, 3);
%e A262729 w in base 7 spells the prime 6598716743.
%t A262729 {b1, b2, b3, b4} = {2, 3, 5, 7}; z = 10000000;
%t A262729 Select[Prime[Range[z]],
%t A262729 PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
%t A262729 PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
%t A262729 PrimeQ[FromDigits[IntegerDigits[#, b1], b4]] &&
%t A262729 PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &&
%t A262729 PrimeQ[FromDigits[IntegerDigits[#, b2], b4]] &&
%t A262729 PrimeQ[FromDigits[IntegerDigits[#, b3], b4]] &]
%t A262729 (* _Peter J. C. Moses_, Sep 27 2015 *)
%Y A262729 Cf. A000040, A262727, A262728.
%K A262729 nonn,base
%O A262729 1,1
%A A262729 _Clark Kimberling_, Oct 03 2015
%E A262729 a(4)-a(22) from _Hiroaki Yamanouchi_, Oct 25 2015