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A217116 Greatest number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

%I A217116 #11 Jul 16 2015 22:26:21
%S A217116 23,839,5039,6983,7127,42743,45863,46199,275183,279143,277199,1088531,
%T A217116 1674863,1651103,1674859,6713711,9906599,10045559,10072943,39190247,
%U A217116 40278647,60273359,60295079,60294239,60437659,241671887,342609527,359245007,361640159,362625959
%N A217116 Greatest number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).
%C A217116 The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 6^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-6 representation 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. This proves the statement of existence. Proof of finiteness: Each 3-digit base-6 number has at least 1 nonprime substring. Hence, each 3(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 6^(3n+2) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.
%H A217116 Hieronymus Fischer, <a href="/A217116/b217116.txt">Table of n, a(n) for n = 0..55</a>
%F A217116 a(n) >= A217106(n).
%F A217116 a(n) >= A217306(A000217(num_digits_6(a(n)))-n), where num_digits_6(x) is the number of digits of the base-6 representation of x.
%F A217116 a(n) <= 6^min(n+3, 7*floor((n+7)/8)).
%F A217116 a(n) <= 216*6^n.
%F A217116 a(n+m+1) >= 6*a(n), where m := floor(log_6(a(n))) + 1.
%e A217116 (0) = 23, since 23 = 35_6 (base-6) is the greatest number with zero nonprime substrings in base-6 representation.
%e A217116 a(1) = 839 = 3515_6 has 1 nonprime substring in base-6 representation (= 1). All the other base-6 substrings (3, 5, 15_6=11, 35_6=23, 51_6=31, 351_6=139, 515_6=191 and 3515_6=839) are prime substrings. 839 is the greatest number with 1 nonprime substring.
%e A217116 a(2) = 5039 = 35155_6 has 2 nonprime substrings in base-6 representation (1 and 55_6=35), and there is no greater number with 2 nonprime substrings in base-6 representation.
%e A217116 a(5) = 42743 = 525515_6 has 5 nonprime substrings in base-6 representation, these are 1, 52_6=32, 55_6=35, 5515_6=1271 and 52551_6=7123, and there is no greater number with 5 nonprime substrings in base-6 representation.
%Y A217116 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217116 Cf. A035244, A079397, A213300 - A213321.
%Y A217116 Cf. A217102 - A217109, A217112 - A217119.
%Y A217116 Cf. A217302 - A217309.
%K A217116 base,nonn
%O A217116 0,1
%A A217116 _Hieronymus Fischer_, Dec 20 2012