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%I #13 Jul 16 2015 22:24:46
%S 2,1,12,9,83,84,81,748,740,731,729,6653,6581,6563,6564,6561,59222,
%T 59069,59068,59051,59052,59049,531614,531569,531464,531460,531452,
%U 531443,531441,4784122,4783142,4783147,4783070,4782989,4782971,4782972,4782969,43048283
%N Minimal 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. Proof: Define m(n):=2*sum_{j=i..k} 9^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-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, which proves the statement.
%C If p is a number with k prime substrings and d digits (in base-9 representation), m>=d, than b := p*9^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.
%H Hieronymus Fischer, <a href="/A217109/b217109.txt">Table of n, a(n) for n = 0..210</a>
%F a(n) >= 9^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
%F a(A000217(n)) = 9^(n-1), n>0.
%F a(A000217(n)-k) >= 9^(n-1) + k, 0<=k<n, n>0.
%F a(A000217(n)-k) = 9^(n-1) + p, where p is the minimal number >= 0 such that 9^(n-1) + p, has k prime substrings in base-9 representation, 0<=k<n, n>0.
%e a(0) = 2, since 2 = 2_9 is the least number with zero nonprime substrings in base-9 representation.
%e a(1) = 1, since 1 = 1_9 is the least number with 1 nonprime substring in base-9 representation.
%e a(2) = 12, since 12 = 13_9 is the least number with 2 nonprime substrings in base-9 representation (1 and 13).
%e a(3) = 9, since 9 = 10_9 is the least number with 3 nonprime substrings in base-9 representation (0, 1 and 10).
%e a(4) = 83, since 83 = 102_9 is the least number with 4 nonprime substrings in base-9 representation, these are 0, 1, 10, and 02 (remember, that substrings with leading zeros are considered to be nonprime).
%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y Cf. A035244, A079397, A213300-A213321.
%Y Cf. A217102-A217109.
%Y Cf. A217302-A217309.
%K nonn,base
%O 0,1
%A _Hieronymus Fischer_, Dec 12 2012
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