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%I #11 Oct 24 2014 04:14:58
%S 1,2,7,11,23,43,93,151,239,373,479,727,1495,2015,2775,5591,6133,7919,
%T 12271,22367,24303,30431,48991,89527,95607,98143,129887,357883,358111,
%U 382431,744797,519551,1431007,1432447,1556319,2457439
%N Minimal natural number (in decimal representation) with n prime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).
%C The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=4*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 4^j = 2*(4^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222, …,for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-4 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.
%C No term is divisible by 4. a(1) = 2 is the only even term.
%H Hieronymus Fischer, <a href="/A217304/b217304.txt">Table of n, a(n) for n = 0..77</a>
%F a(n) > 4^floor(sqrt(8*n-7)-1)/2), for n>0.
%F a(n) <= 2*(4^n - 1)/3, n>0.
%F a(n+1) <= 4*a(n) + 2.
%e a(1) = 2 = 2_4, since 2 is the least number with 1 prime substring in base-4 representation.
%e a(2) = 7 = 13_4, since 7 is the least number with 2 prime substrings in base-4 representation (3_4=3 and 13_4=7).
%e a(3) = 11 = 23_4, since 11 is the least number with 3 prime substrings in base-4 representation (2_4, 3_4, and 23_4).
%e a(5) = 43 = 223_4, since 43 is the least number with 5 prime substrings in base-4 representation (2 times 2_4, 3_4, 23_4=11, and 223_4=43).
%e a(7) = 151 = 2113_4, since 151 is the least number with 7 prime substrings in base-4 representation (2 times 2_4, 3_4, 11_4=5, 13_4=7, 113_4=23, and 2113_4=151).
%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y Cf. A035244, A079397, A213300-A213321.
%Y Cf. A217302-A217309.
%K nonn,base
%O 0,2
%A _Hieronymus Fischer_, Nov 22 2012
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