cp's OEIS Frontend

A217108 Minimal number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

%I A217108 #15 Jul 16 2015 22:24:27
%S A217108 2,1,10,8,67,66,64,523,525,514,512,4127,4115,4099,4098,4096,32797,
%T A217108 32799,32779,32771,32770,32768,262237,262239,262173,262163,262147,
%U A217108 262146,262144,2097391,2097259,2097211,2097181,2097169,2097163,2097154,2097152,16777695
%N A217108 Minimal number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).
%C A217108 The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 8^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). 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, which proves the statement.
%C A217108 If p is a number with k prime substrings and d digits (in base-8 representation), m>=d, than b := p*8^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.
%H A217108 Hieronymus Fischer, <a href="/A217108/b217108.txt">Table of n, a(n) for n = 0..210</a>
%F A217108 a(n) >= 8^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
%F A217108 a(A000217(n)) = 8^(n-1), n>0.
%F A217108 a(A000217(n)-k) >= 8^(n-1) + k, 0<=k<n, n>0.
%F A217108 a(A000217(n)-k) = 8^(n-1) + p, where p is the minimal number >= 0 such that 8^(n-1) + p, has k prime substrings in base-8 representation, 0<=k<n, n>0.
%e A217108 a(0) = 2, since 2 = 2_8 is the least number with zero nonprime substrings in base-8 representation.
%e A217108 a(1) = 1, since 1 = 1_8 is the least number with 1 nonprime substring in base-8 representation.
%e A217108 a(2) = 10, since 10 = 12_8 is the least number with 2 nonprime substrings in base-8 representation (1 and 12).
%e A217108 a(3) = 8, since 8 = 10_8 is the least number with 3 nonprime substrings in base-8 representation (0, 1 and 10).
%e A217108 a(4) = 67, since 67 = 103_8 is the least number with 4 nonprime substrings in base-8 representation, these are 0, 1, 10, and 03 (remember, that substrings with leading zeros are considered to be nonprime).
%Y A217108 Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y A217108 Cf. A035244, A079397, A213300-A213321.
%Y A217108 Cf. A217102-A217109.
%Y A217108 Cf. A217302-A217309.
%K A217108 nonn,base
%O A217108 0,1
%A A217108 _Hieronymus Fischer_, Dec 12 2012