cp's OEIS Frontend

A217309 Minimal natural number (in decimal representation) with n prime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

%I A217309 #11 Oct 24 2014 04:15:33
%S A217309 1,2,11,23,101,173,902,1562,1559,8120,14032,14033,73082,126290,604523,
%T A217309 657743,723269,1136684,5918933,5972147,10227787,25051529,53276231,
%U A217309 54333278,92071913,441753767,479669051,483743986,828662228,3971590751,4315446629
%N A217309 Minimal natural number (in decimal representation) with n prime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).
%C A217309 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):=9*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 9^j = (9^n - 1)/4 or m(n)=1, 2, 22, 222, 2222, 22222, …, (in base-9) 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-9 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 a prime number.
%C A217309 No term is divisible by 9.
%H A217309 Hieronymus Fischer, <a href="/A217309/b217309.txt">Table of n, a(n) for n = 0..32</a>
%F A217309 a(n) > 9^floor(sqrt(8*n-7)-1)/2), for n>0.
%F A217309 a(n) <= (9^n - 1)/4, n>0.
%F A217309 a(n+1) <= 9*a(n)+3.
%e A217309 a(1) = 2 = 2_9, since 2 is the least number with 1 prime substring in base-9 representation.
%e A217309 a(2) = 11 = 12_9, since 11 is the least number with 2 prime substrings in base-9 representation (2_9 and 12_9).
%e A217309 a(3) = 23 = 25_9, since 23 is the least number with 3 prime substrings in base-9 representation (2_9, 3_9, and 23_9).
%e A217309 a(4) = 101 = 122_9, since 101 is the least number with 4 prime substrings in base-9 representation (2 times 2_9, 12_9=11, and 122_9=101).
%e A217309 a(7) = 1562 = 2125_9, since 1562 is the least number with 7 prime substrings in base-9 representation (2 times 2_9, 5_9, 12_9=11, 21_9=19, 25_9=23, and 212_9=173).
%Y A217309 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217309 Cf. A035244, A079397, A213300-A213321.
%Y A217309 Cf. A217302-A217308.
%K A217309 nonn,base
%O A217309 0,2
%A A217309 _Hieronymus Fischer_, Nov 22 2012