cp's OEIS Frontend

A217305 Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

%I A217305 #13 Jun 02 2015 08:30:40
%S A217305 1,2,7,13,37,88,67,192,317,932,942,1567,4663,4692,8442,23317,23442,
%T A217305 36067,102217,114192,180337,192317,511087,901682,582942,2495443,
%U A217305 2555436,2536067,5289942,12321061,12680337,12301692,26461592,61508461,61508462,63885918
%N A217305 Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).
%C A217305 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):=5*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 5^j = (5^n - 1)/2 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-5) 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-5 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 A217305 No term is divisible by 5.
%H A217305 Hieronymus Fischer, <a href="/A217305/b217305.txt">Table of n, a(n) for n = 0..53</a>
%F A217305 a(n) > 5^floor(sqrt(8*n-7)-1)/2), for n>0.
%F A217305 a(n) <= (5^n - 1)/2, n>0.
%e A217305 a(1) = 2 = 2_5, since 2 is the least number with 1 prime substring in base-5 representation.
%e A217305 a(2) = 7 = 12_5, since 7 is the least number with 2 prime substrings in base-5 representation (2_5 and 12_5=7).
%e A217305 a(3) = 13 = 23_5, since 13 is the least number with 3 prime substrings in base-5 representation (2_5, 3_5, and 23_5).
%e A217305 a(4) = 37 = 122_5, since 37 is the least number with 4 prime substrings in base-5 representation (2 times 2_5, 12_5=7, and 122_5=37).
%e A217305 a(7) = 192 = 1232_5, since 192 is the least number with 7 prime substrings in base-5 representation (2 times 2_5, 3_5, 12_5=7, 23_5=13, 32_5=17, and 232_5=67).
%Y A217305 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217302-A217309.
%K A217305 nonn,base
%O A217305 0,2
%A A217305 _Hieronymus Fischer_, Nov 22 2012