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Greatest number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).
9

%I #11 Jul 16 2015 22:27:44

%S 47,428,1721,6473,14033,35201,58961,58967,465743,530701,530710,

%T 1733741,4250788,4723108,4776398,25051529,37327196,42450640,42986860,

%U 42987589,42996409,225463817,382055767,382571822,386888308,386888419,387356789

%N Greatest number (in decimal representation) with n nonprime substrings in base-9 representation (substrings with leading zeros are considered to be nonprime).

%C 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} 9^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-9 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-9 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 < 9^(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 Hieronymus Fischer, <a href="/A217119/b217119.txt">Table of n, a(n) for n = 0..40</a>

%F a(n) >= A217109(n).

%F a(n) >= A217309(A000217(num_digits_9(a(n)))-n), where num_digits_9(x)=floor(log_9(x))+1 is the number of digits of the base-9 representation of x.

%F a(n) <= 9^(n+2).

%F a(n) <= 9^min(n+2, 6*floor((n+7)/8)).

%F a(n) <= 9^((3/4)*(n + 3)).

%F a(n+m+1) >= 9*a(n), where m := floor(log_9(a(n))) + 1.

%e a(0) = 47, since 47 = 52_9 (base-9) is the greatest number with zero nonprime substrings in base-9 representation.

%e a(1) = 428 = 525_9 has 1 nonprime substring in base-9 representation (= 525_9). All the other base-9 substrings (2, 5, 5, 25, 52) are prime substrings. 525_9 is the greatest number with 1 nonprime substring.

%e a(2) = 1721 = 2322_9 has 10 substrings in base-9 representation, exactly 2 of them are nonprime substrings (22_9 and 23_3=8), and there is no greater number with 2 nonprime substrings in base-9 representation.

%e a(7) = 58967= 88788_9 has 15 substrings in base-9 representation, exactly 7 of them are nonprime substrings (4-times 8, 2-times 88, and 8788), and there is no greater number with 7 nonprime substrings in base-9 representation.

%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

%Y Cf. A035244, A079397, A213300 - A213321.

%Y Cf. A217102 - A217109, A217112 - A217118.

%Y Cf. A217302 - A217309.

%K nonn,base

%O 0,1

%A _Hieronymus Fischer_, Dec 20 2012