cp's OEIS Frontend

A217114 Greatest number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

%I A217114 #11 Jul 16 2015 22:25:28
%S A217114 11,59,239,251,751,1007,1019,3823,4079,4055,16111,16087,16319,16367,
%T A217114 48991,64351,65263,65269,65471,253919,260959,261079,261847,261871,
%U A217114 916319,1043839,1047391,1044463,1047511,3665279,3140991,4189567,4118519,4177759,4189565,4193239,14661117
%N A217114 Greatest number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).
%C A217114 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} 4^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-4 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-4 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 < 4^(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 A217114 Hieronymus Fischer, <a href="/A217114/b217114.txt">Table of n, a(n) for n = 0..100</a>
%F A217114 a(n) >= A217104(n).
%F A217114 a(n) >= A217304(A000217(A110591(a(n)))-n).
%F A217114 a(n) <= 4^(n+2).
%F A217114 a(n) <= 4^min((n + 6)/2, 9*floor((n+18)/19)).
%F A217114 a(n) <= 64*4^(n/2).
%F A217114 a(n+m+1) >= 4*a(n), where m := floor(log_4(a(n))) + 1.
%e A217114 a(0) = 11, since 11 = 23_4 (base-4) is the greatest number with zero nonprime substrings in base-4 representation.
%e A217114 a(1) = 59 = 323_4 has 6 substrings in base-4 representation (2, 3, 3, 23, 32 and 323), only 32_4=14 is a nonprime substring. 59 is the greatest such number with 1 nonprime substring.
%e A217114 a(2) = 239 = 3233_4 has 10 substrings in base-4 representation (2, 3, 3, 23, 32, 323, 233 and 3233), exactly 2 of them are nonprime substrings (32_4=14 and 33_4=15), and there is no greater number with 2 nonprime substrings in base-4 representation.
%e A217114 a(11) = 16087 = 3323113_4 has 28 substrings in base-4 representation. The base-4 nonprime substrings are 1, 1, 32, 33, 231, 332, 3113, 3231, 32311, 33321 and 323113. There is no greater number with 11 nonprime substrings in base-4 representation.
%Y A217114 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217114 Cf. A035244, A079397, A213300 - A213321.
%Y A217114 Cf. A217102 - A217109, A217112 - A217119.
%Y A217114 Cf. A217302 - A217309.
%K A217114 nonn,base
%O A217114 0,1
%A A217114 _Hieronymus Fischer_, Dec 20 2012