cp's OEIS Frontend

A217113 Greatest number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

%I A217113 #12 Jul 16 2015 22:25:07
%S A217113 2,23,71,26,77,233,239,719,701,647,725,2159,2177,2158,2157,5822,5741,
%T A217113 6551,6476,6532,6531,18944,19436,19655,19601,19673,19653,58310,58309,
%U A217113 58316,58967,59021,58964,157211,157217,174950,176408,176407,176903,177065,177064,471653,511511
%N A217113 Greatest number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).
%C A217113 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} 3^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-3 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 2-digit base-3 number has at least 1 nonprime substring. Hence, each 2(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 3^(2n+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 A217113 Hieronymus Fischer, <a href="/A217113/b217113.txt">Table of n, a(n) for n = 0..250</a>
%F A217113 a(n) >= A217103(n).
%F A217113 a(n) >= A217303(A000217(A081604(a(n)))-n).
%F A217113 Example: a(12)=2177=2222122_3, A000217(A081604(2177))=28, hence a(12)>=A217303(28-12)=1934.
%F A217113 a(n) <= 3^min(n + 2, 5*floor((n+4)/5)).
%F A217113 a(n) <= 3^(n + 2).
%F A217113 a(n) <= 3^min((n + 11)/3, 11*floor((n+32)/33)).
%F A217113 a(n) <= 3^((1/3)*(n + 11)).
%F A217113 With m := floor(log_3(a(n))) + 1:
%F A217113 a(n+m+1) >= 3*a(n), if a(n)!=1 (mod 3).
%F A217113 a(n+m) >= 3*a(n), if a(n)=1 (mod 3).
%e A217113 a(0) = 2, since 2 = 2_3 (base-3) is the greatest number with zero nonprime substrings in base-3 representation.
%e A217113 a(1) = 23 = 212_3 has 1 substring in base-3 representation (= 1). All the other base-3 substrings (2, 2, 21, 12, 212) are prime substrings. 23 is the greatest number with 1 nonprime substring.
%e A217113 a(2) = 71 = 2122_3 has 10 substrings in base-3 representation (1, 2, 2, 2, 12, 21, 22, 122, 212, 2122), exactly 2 of them are nonprime substrings (1 and 22_3=8), and there is no greater number with 2 nonprime substrings in base-3 representation.
%e A217113 a(3) = 26 = 222_3 has 6 substrings in base-3 representation, only 3 of them are prime substrings (2, 2, 2) which implies that exactly 3 substrings must be nonprime, and there is no greater number with 3 nonprime substrings in base-3 representation.
%Y A217113 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217113 Cf. A035244, A079397, A213300 - A213321.
%Y A217113 Cf. A217102 - A217109, A217112 - A217119.
%Y A217113 Cf. A217302 - A217309.
%K A217113 nonn,base
%O A217113 0,1
%A A217113 _Hieronymus Fischer_, Dec 20 2012