cp's OEIS Frontend

A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

%I A217115 #12 Jul 16 2015 22:25:50
%S A217115 67,88,442,567,2213,2837,3067,11068,14713,15337,15338,57943,73568,
%T A217115 77213,76697,289717,280338,370443,386068,386587,389713,1852217,
%U A217115 1524067,1898442,1930342,1932943,1948568,7242943,9261088,9664717,9586567,9654712,9710942,9742849,46305443
%N A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).
%C A217115 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} 5^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-5 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-5 number has at least 2 nonprime substrings. Hence, each m := 4*floor((n+2)/2)-digit number has at least 2*(m/4) = 2*floor((n+2)/2) >= n+1 nonprime substrings. Consequently, there is a boundary b < 5^(m-1) 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 A217115 Hieronymus Fischer, <a href="/A217115/b217115.txt">Table of n, a(n) for n = 0..70</a>
%F A217115 a(n) >= A217105(n).
%F A217115 a(n) >= A217305(A000217(A110592(a(n)))-n).
%F A217115 a(n) <= 5^(n+3).
%F A217115 a(n) <= 5^(4*floor(n/2)), n>1.
%F A217115 a(n) <= 5^min((n + 6)/2, 9*floor((n+20)/21)).
%F A217115 a(n) <= 125*5^(n/2).
%F A217115 With m := floor(log_5(a(n))) + 1:
%F A217115 a(n+m+1) >= 5*a(n), if a(n)!=1 (mod 5).
%F A217115 a(n+m) >= 5*a(n), if a(n)=1 (mod 5).
%e A217115 a(0) = 67, since 67 = 232_5 (base-5) is the greatest number with zero nonprime substrings in base-5 representation.
%e A217115 a(1) = 88 = 323_5 has 6 substrings in base-5 representation (2, 2, 3, 23, 32, 323), the only nonprime substring is 323_5. 88 is the greatest number with 1 nonprime substring.
%e A217115 a(2) = 442 = 3232_5 has 10 substrings in base-5 representation (2, 2, 3, 3, 23, 32, 32, 232, 323 and 3232), exactly 2 of them are nonprime substrings (323_5=88 and 3232_5=442), and there is no greater number with 2 nonprime substrings in base-5 representation.
%e A217115 a(5) = 2837 = 42322_5 has 5 nonprime substrings in base-5 representation, these are 4, 22, 42, 322 and 4232, all the other substrings are prime. There is no greater number with 5 nonprime substrings in base-5 representation.
%Y A217115 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217115 Cf. A035244, A079397, A213300 - A213321.
%Y A217115 Cf. A217102 - A217109, A217112 - A217119.
%Y A217115 Cf. A217302 - A217309.
%K A217115 nonn,base
%O A217115 0,1
%A A217115 _Hieronymus Fischer_, Dec 20 2012