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Minimal number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).
2

%I #13 Jul 16 2015 22:23:16

%S 2,1,7,6,41,37,36,223,224,218,216,1319,1307,1301,1297,1296,7829,7793,

%T 7787,7783,7778,7776,46703,46709,46679,46673,46663,46658,46656,280205,

%U 280075,279983,279979,279949,279941,279938,279936,1679879,1679807,1679699,1679669

%N Minimal number (in decimal representation) with n nonprime substrings in base-6 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} 6^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-6 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-6 representation), m>=d, than b := p*6^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

%H Hieronymus Fischer, <a href="/A217106/b217106.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) >= 6^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).

%F a(A000217(n)) = 6^(n-1), n>0.

%F a(A000217(n)-k) >= 6^(n-1) + k, 0<=k<n, n>0.

%F a(A000217(n)-k) = 6^(n-1) + p, where p is the minimal number >= 0 such that 6^(n-1) + p, has k prime substrings in base-6 representation, 0<=k<n, n>0.

%e a(0) = 2, since 2 = 2_6 is the least number with zero nonprime substrings in base-6 representation.

%e a(1) = 1, since 1 = 1_6 is the least number with 1 nonprime substring in base-6 representation.

%e a(2) = 7, since 7 = 11_6 is the least number with 2 nonprime substrings in base-6 representation (1 and 1).

%e a(3) = 6, since 6 = 10_6 is the least number with 3 nonprime substrings in base-6 representation (0, 1 and 10).

%e a(4) = 41, since 41 = 105_6 is the least number with 4 nonprime substrings in base-6 representation, these are 0, 1, 10, and 05 (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