|
%I #15 Jul 16 2015 22:22:52
%S 2,1,5,6,27,25,34,127,128,125,170,636,632,627,625,850,3162,3137,3132,
%T 3127,3125,4250,15686,15661,15638,15632,15627,15625,21250,78192,78163,
%U 78162,78137,78132,78127,78125,106250,390818,390692,390686,390662,390638,390632
%N Minimal number (in decimal representation) with n nonprime substrings in base-5 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} 5^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-5 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-5 representation), p != 1 (mod 5), m>=d, than b := p*5^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.
%H Hieronymus Fischer, <a href="/A217105/b217105.txt">Table of n, a(n) for n = 0..351</a>
%F a(n) >= 5^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n=1 or n+1 is a triangular number (cf. A000217).
%F a(A000217(n)-1) = 5^(n-1), n>1.
%F a(A000217(n)) = floor(34 * 5^(n-3)), n>0.
%F a(A000217(n)) = 114000...000_5 (with n digits), n>0.
%F a(A000217(n)-k) >= 5^(n-1) + k-1, 1<=k<=n, n>1.
%F a(A000217(n)-k) = 5^(n-1) + p, where p is the minimal number >= 0 such that 5^(n-1) + p, has k prime substrings in base-5 representation, 1<=k<=n, n>1.
%e a(0) = 2, since 2 = 2_5 is the least number with zero nonprime substrings in base-4 representation.
%e a(1) = 1, since 1 = 1_5 is the least number with 1 nonprime substring in base-5 representation.
%e a(2) = 5, since 5 = 10_5 is the least number with 2 nonprime substrings in base-5 representation (0 and 1).
%e a(3) = 6, since 6 = 11_5 is the least number with 3 nonprime substrings in base-5 representation (2-times 1 and 11).
%e a(4) = 27, since 27 = 102_5 is the least number with 4 nonprime substrings in base-5 representation, these are 0, 1, 02, and 102 (remember, that substrings with leading zeros are considered to be nonprime).
%e a(6) = 34, since 34 = 114_5 is the least number with 6 nonprime substrings in base-5 representation, these are 1, 1, 4, 11, 14, and 114.
%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
%Y Cf. A035244, A079397.
%Y Cf. A217102-A217109, A217112-A217119.
%Y Cf. A217302-A217309.
%K nonn,base
%O 0,1
%A _Hieronymus Fischer_, Dec 12 2012
|
