cp's OEIS Frontend

A213320 Numbers such that the number of nonprime substrings equals the number of digits (substrings with leading zeros are considered to be nonprime).

%I A213320 #20 Jul 16 2015 22:16:23
%S A213320 1,4,6,8,9,11,12,15,19,20,21,24,26,28,30,34,36,38,39,41,42,45,50,51,
%T A213320 54,56,58,61,62,63,65,70,74,76,78,82,85,87,89,92,93,95,117,123,127,
%U A213320 132,133,135,139,153,157,167,171,172,175
%N A213320 Numbers such that the number of nonprime substrings equals the number of digits (substrings with leading zeros are considered to be nonprime).
%C A213320 Also numbers such that the number of prime substrings is A000217(m-1) = m(m-1)/2, where m is the number of digits.
%C A213320 The sequence is finite. Proof: Let p be a number >= 10^17 and let m = 9k+j be the number of digits of p, where k = floor(m/9) >= 2 and j = m mod 9. Since each 9-digit number has at least 15 nonprime substrings, it follows that p has at least 15k = 9k + 6k > 9k + j = m nonprime substrings (since 6k >= 12> j for k >= 2). Consequently, no number >= 10^17 can be a term of the sequence.
%C A213320 The last term is a(858)=3733739. Proof: Each 9-digit number has at least 15 nonprime substrings, thus, the numbers 10^8 <= p < 10^14 also have at least 15 nonprime substrings and therefore cannot be terms of the sequence. Same is true for numbers 10^14 <= p < 10^17 since each 6-digit number has at least 4 nonprime substrings, and thus each number with >= 15 digits has at least 15+4 = 19 nonprime substrings. Since each 8-digit number has at least 10 nonprime substrings, it follows that the last term of the sequence must be less than 10^7. By direct search we find a(858) = 3733739.
%H A213320 Hieronymus Fischer, <a href="/A213320/b213320.txt">Table of n, a(n) for n = 1..858</a> (full sequence)
%e A213320 a(1) = 1, since 1 has 1 nonprime substrings.
%e A213320 a(43) = 117, since 117 has 3 digits and also 3 nonprime substrings (1, 1, 117).
%Y A213320 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A213320 Cf. A035244, A079307, A213300 - A213321.
%K A213320 nonn,fini,full,base
%O A213320 1,2
%A A213320 _Hieronymus Fischer_, Aug 26 2012
%E A213320 Typo in example corrected, _Hieronymus Fischer_, Sep 11 2012