%I #14 Jul 16 2015 22:15:38

%S 2,1,10,14,101,104,144,1001,1014,1044,1444,10010,10014,10144,10444,

%T 14444,100120,100104,100144,101444,104444,144444,1000144,1001040,

%U 1001044,1001444,1014444,1044444,1444444,10001044,10001444,10010404,10010444,10014444,10144444,10444444,14444444,100010404,100010444,100014444,100104044,100104444,100144444,101444444,104444444,144444444

%N Smallest number with n nonprime substrings (Version 3: substrings with leading zeros are counted as nonprime if the corresponding number is not a prime).

%C The sequence is well defined since for each n >= 0 there is a number with n nonprime substrings.

%C Different from A213303, first difference is at a(16).

%H Hieronymus Fischer, <a href="/A213304/b213304.txt">Table of n, a(n) for n = 0..100</a>

%F a(m(m+1)/2) = (13*10^(m-1)-4)/9, m>0.

%F With b(n):=floor((sqrt(8*n-7)-1)/2):

%F a(n) > 10^b(n), for n>2, a(n) = 10^b(n) for n=1,2.

%F a(n) >= 10^b(n)+4*10^(n-1-b(n)(b(n)+1)/2)-1)/9, equality holds if n or n+1 is a triangular number > 0 (cf. A000217).

%F a(n) >= A213303(n).

%F a(n) <= A213307(n).

%e a(0)=2, since 2 is the least number with zero nonprime substrings.

%e a(1)=1, since 1 has 1 nonprime substrings.

%e a(2)=10, since 10 is the least number with 2 nonprime substrings, these are 1 and 10 ('0' will not be counted).

%e a(3)=14, since 14 is the least number with 3 nonprime substrings, these are 1 and 4 and 14. 10, 11 and 12 only have 2 such substrings.

%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

%Y Cf. A035244, A079307, A213300 - A213321.

%K nonn,base

%O 0,1

%A _Hieronymus Fischer_, Aug 26 2012