cp's OEIS Frontend

A213302 Smallest number with n nonprime substrings (Version 1: substrings with leading zeros are considered to be nonprime).

%I A213302 #10 Oct 23 2014 14:12:55
%S A213302 2,1,11,10,103,101,100,1017,1011,1002,1000,10037,10023,10007,10002,
%T A213302 10000,100137,100073,100023,100003,100002,100000,1000313,1000037,
%U A213302 1000033,1000023,1000003,1000002,1000000,10000337,10000223,10000137,10000037,10000023,10000013,10000002,10000000,100001733
%N A213302 Smallest number with n nonprime substrings (Version 1: substrings with leading zeros are considered to be nonprime).
%C A213302 The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n)=2*sum_{j=i..k} 10^j, where k=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,… the m(n) are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, ... . m(n) has k+1 digits and (k-i+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i=(k*(k+1)/2)+i=n, which proves the statement.
%C A213302 The 3 versions according to A213302 - A213304 are quite different. Example: 1002 has 9 nonprime substrings in version 1 (0, 0, 00, 02, 002, 1, 10 100, 1002), in version 2 there are 6 nonprime substrings (02, 002, 1, 10, 100, 1002) and there are 4 nonprime substrings in version 3 (1, 10, 100, 1002).
%H A213302 Hieronymus Fischer, <a href="/A213302/b213302.txt">Table of n, a(n) for n = 0..200</a>
%F A213302 a(n) >= 10^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number > 0 (cf. A000217).
%F A213302 a(A000217(n)) = 10^(n-1), n>0.
%F A213302 a(A000217(n)-k) >= 10^(n-1)+k, n>0, 0<=k<n.
%F A213302 a(A000217(n)-1) = 10^(n-1)+2, n>3, provided 10^(n-1)+1 is not a prime (which is proved to be true for all n-1 <= 50000 (cf. A185121) except n-1=16384 and is generally true for n-1 unequal to a power of 2).
%F A213302 a(A000217(n)-k) = 10^(n-1)+p, where p is the minimal number such that 10^(n-1) + p, has k prime substrings, n>0, 0<=k<n.
%F A213302 Min(a(A000217(n)-k-i), 0<=i<=m) <= 10^(n-1)+p, where p is the minimal number with k prime substrings and m is the number of digits of p, and k+m<n.
%F A213302 Min(a(A000217(n)-k-i), 0<=i<=A055642(A035244(k)) <= 10^(n-1)+A035244(k).
%F A213302 a(A000217(n)-k) <= 10^(n-1)+max(p(i), k<=i<=k+m), where p(i) is the minimal number with i prime substrings and m is the number of digits of p(i), and k+m<n.
%F A213302 a(A000217(n)-k) <= 10^(n-1)+max(A035244(i), k<=i<=k+ A055642(i).
%F A213302 a(n) <= A213305(n).
%e A213302 a(0)=2, since 2 is the least number with zero nonprime substrings.
%e A213302 a(1)=1, since 1 has 1 nonprime substrings.
%e A213302 a(2)=11, since 11 is the least number with 2 nonprime substrings.
%e A213302 a(3)=10, since 10 is the least number with 3 nonprime substrings, these are 1, 0 and 10 (‘0’ will be counted).
%Y A213302 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A213302 Cf. A035244, A079307, A213300 - A213321.
%K A213302 nonn,base
%O A213302 0,1
%A A213302 _Hieronymus Fischer_, Aug 26 2012