cp's OEIS Frontend

A217306 Minimal natural number (in decimal representation) with n prime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

%I A217306 #12 Oct 24 2014 04:15:13
%S A217306 1,2,11,17,47,83,269,263,479,839,1559,1579,2999,5039,9355,9479,14759,
%T A217306 56131,56135,61343,56879,336791,341351,336815,341279,341275,2020727,
%U A217306 2020895,2047651,2020891,4055159,12098587,12125347,12285907,15737755,19128523,39190247
%N A217306 Minimal natural number (in decimal representation) with n prime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).
%C A217306 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):=6*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 6^j = 2*(6^n - 1)/5 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-6) 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-6 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 A217306 No term is divisible by 6.
%H A217306 Hieronymus Fischer, <a href="/A217306/b217306.txt">Table of n, a(n) for n = 0..60</a>
%F A217306 a(n) > 6^floor(sqrt(8*n-7)-1)/2), for n>0.
%F A217306 a(n) <= 2*(6^n - 1)/5, n>0.
%F A217306 a(n+1) <= 6*a(n)+2.
%e A217306 a(1) = 2 = 2_6, since 2 is the least number with 1 prime substring in base-6 representation.
%e A217306 a(2) = 11 = 15_6, since 11 is the least number with 2 prime substrings in base-6 representation (5_6=5 and 15_6=11).
%e A217306 a(3) = 17 = 25_6, since 17 is the least number with 3 prime substrings in base-6 representation (2_6, 5_6, and 25_6).
%e A217306 a(4) = 47 = 115_6, since 47 is the least number with 4 prime substrings in base-6 representation (5_6, 11_6=7, 15_6=11, and 115_6=47).
%e A217306 a(8) = 479 = 2115_6, since 479 is the least number with 8 prime substrings in base-6 representation (2_6, 5_6, 11_6=7, 15_6=11, 21_6=13, 115_6=47, 211_6=79, and 2115_6=479).
%Y A217306 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217306 Cf. A035244, A079397, A213300-A213321.
%Y A217306 Cf. A217302-A217309.
%K A217306 nonn,base
%O A217306 0,2
%A A217306 _Hieronymus Fischer_, Nov 22 2012