cp's OEIS Frontend

A217118 Greatest number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

%I A217118 #11 Jul 16 2015 22:27:26
%S A217118 491,3933,24303,32603,188143,253789,261117,1555423,2030319,2088797,
%T A217118 2088943,16185163,16710383,16710381,16768991,99606365,129884143,
%U A217118 133683069,134150015,134209503,770611067,1039073149,1069408239,1073209071,1073209083,1073676029,5065578363
%N A217118 Greatest number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).
%C A217118 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} 8^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-8 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 4-digit base-8 number has at least 1 nonprime substring. Hence, each 4(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 8^(4n+3) 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 A217118 Hieronymus Fischer, <a href="/A217118/b217118.txt">Table of n, a(n) for n = 0..50</a>
%F A217118 a(n) >= A217108(n).
%F A217118 a(n) >= A217308(A000217(num_digits_8(a(n)))-n), where num_digits_8(x) is the number of digits of the base-8 representation of x.
%F A217118 a(n) <= 8^min(n+3, 7*floor((n+6)/7)).
%F A217118 a(n) <= 512*8^n.
%F A217118 a(n+m+1) >= 8*a(n), where m := floor(log_8(a(n))) + 1.
%e A217118 a(0) = 491, since 491 = 753_8 (base-8) is the greatest number with zero nonprime substrings in base-8 representation.
%e A217118 a(1) = 3933 = 7535_8 has 1 nonprime substring in base-8 representation (=7535_8). All the other base-8 substrings are prime substrings. 3933 is the greatest such number with 1 nonprime substring.
%e A217118 a(2) = 24303 = 57357_8 has 15 substrings in base-8 representation, exactly 2 of them are nonprime substrings (57357_8 and 735_8), and there is no greater number with 2 nonprime substrings in base-3 representation.
%e A217118 a(3) = 32603 = 77533_8 has 15 substrings in base-8 representation, only 3 of them are nonprime substrings (33_8, 77_8, and 7753_8), and there is no greater number with 3 nonprime substrings in base-8 representation.
%Y A217118 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217118 Cf. A035244, A079397, A213300 - A213321.
%Y A217118 Cf. A217102 - A217109, A217112 - A217119.
%Y A217118 Cf. A217302 - A217309.
%K A217118 nonn,base
%O A217118 0,1
%A A217118 _Hieronymus Fischer_, Dec 20 2012