A217308 - OEIS

%I #12 Oct 24 2014 04:15:26

%S 1,2,11,19,83,107,157,669,751,1259,4957,6879,6011,14303,47071,48093,

%T 65371,188143,327515,440287,384751,1029883,2604783,2948955,3602299,

%U 6946651,20304733,23846747,23937003,23723867,57278299,167689071,175479547,191496027,233824091

%N Minimal natural number (in decimal representation) with n prime substrings in base-8 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):=8*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 8^j = 2*(8^n - 1)/7 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-8) 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-8 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 8.

%H Hieronymus Fischer, <a href="/A217308/b217308.txt">Table of n, a(n) for n = 0..45</a>

%F a(n) > 8^floor(sqrt(8*n-7)-1)/2), for n>0.

%F a(n) <= 2*(8^n - 1)/7, n>0.

%F a(n+1) <= 8*a(n)+2.

%e a(1) = 2 = 2_8, since 2 is the least number with 1 prime substring in base-8 representation.

%e a(2) = 11 = 13_8, since 11 is the least number with 2 prime substrings in base-8 representation (3_8 and 13_8).

%e a(3) = 19 = 23_8, since 19 is the least number with 3 prime substrings in base-8 representation (2_8, 3_8, and 23_8).

%e a(4) = 83 = 123_8, since 83 is the least number with 4 prime substrings in base-8 representation (2_8, 3_8, 23_8=19, and 123_8=83).

%e a(8) = 751 = 1357_8, since 751 is the least number with 8 prime substrings in base-8 representation (3_8, 5_8, 7_8, 13_8=11, 35_8=29, 57_8=47, 357_8=239, and 1357_8=751).

%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