cp's OEIS Frontend

A217303 Minimal natural number (in decimal representation) with n prime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

%I A217303 #11 Oct 24 2014 04:14:52
%S A217303 1,2,5,11,17,23,50,104,71,152,215,395,476,701,719,1367,1934,1448,4127,
%T A217303 4121,4346,5822,12302,12383,17468,25505,32066,39113,51749,91040,
%U A217303 111509,110798,117359,157211,332396,334358,465092,333791,819386,865232,1001375,1396673
%N A217303 Minimal natural number (in decimal representation) with n prime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).
%C A217303 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):=3*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 3^j = 3^n - 1 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-3 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 A217303 No term is divisible by 3.
%H A217303 Hieronymus Fischer, <a href="/A217303/b217303.txt">Table of n, a(n) for n = 0..110</a>
%F A217303 a(n) > 3^floor(sqrt(8*n+1)-1)/2), for n>1.
%F A217303 a(n) <= 3^n - 1.
%F A217303 a(n+1) <= 3a(n)+2.
%e A217303 a(1) = 2 = 2_3, since 2 is the least number with 1 prime substring in base-3 representation.
%e A217303 a(2) = 5 = 12_3, since 5 is the least number with 2 prime substrings in base-3 representation (2_3 and 12_3).
%e A217303 a(3) = 11 = 102_3, since 11 is the least number with 3 prime substrings in base-3 representation (2_3, 10_3, and 102_3).
%e A217303 a(5) = 23 = 212_3, since 23 is the least number with 5 prime substrings in base-3 representation (2 times 2_3, 12_3=5, 21_3=19, and 212_3=23).
%e A217303 a(7) = 104 = 10212_3, since 104 is the least number with 7 prime substrings in base-3 representation (2 times 2_3, 10_3=3, 12_3=5, 21_3=19, 102_3=11, and 212_3=23).
%Y A217303 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217303 Cf. A035244, A079397, A213300-A213321.
%Y A217303 Cf. A217302-A217309.
%K A217303 nonn,base
%O A217303 0,2
%A A217303 _Hieronymus Fischer_, Nov 22 2012