A217102 - OEIS

%I #16 Oct 31 2024 06:47:39

%S 1,2,7,5,4,11,10,12,8,22,21,19,17,16,60,39,37,34,36,32,83,71,74,69,67,

%T 66,64,143,139,141,135,134,131,130,128,283,271,269,263,267,262,261,

%U 257,256,541,539,527,526,523,533,519,514,516,512,1055,1053,1047,1067

%N Minimal number (in decimal representation) with n nonprime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

%C There are no numbers with zero nonprime substrings in binary representation. For all bases > 2 there is always a number (=2) with zero nonprime substrings (Cf. A217103-A217109, A213302).

%C If p is a number with k prime substrings and d digits (in binary representation), p even, m>=d, than b := p*2^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

%H Hieronymus Fischer, <a href="/A217102/b217102.txt">Table of n, a(n) for n = 1..2015</a>

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

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

%F a(A000217(n)-1) = 2^(n-1), n>1.

%F a(A000217(n)-k) >= 2^(n-1) + k-1, 1<=k<=n, n>1.

%F a(A000217(n)-k) = 2^(n-1) + p, where p is the minimal number >= 0 such that 2^(n-1) + p, has k prime substrings in binary representation, 1<=k<=n, n>1.

%e a(1) = 1, since 1 = 1_2 is the least number with 1 nonprime substring in binary representation.

%e a(2) = 2, since 2 = 10_2 is the least number with 2 nonprime substrings in binary representation (0 and 1).

%e a(3) = 7, since 7 = 111_2 is the least number with 3 nonprime substrings in binary representation (3-times 1, the prime substrings are 2-times 11 and 111).

%e a(10) = 22, since 22 = 10110_2 is the least number with 10 nonprime substrings in binary representation, these are 0, 0, 1, 1, 1, 01, 011, 110, 0110 and 10110 (remember, that substrings with leading zeros are considered to be nonprime).

%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217103-A217109, A217302-A217309.

%K nonn,base

%O 1,2

%A _Hieronymus Fischer_, Dec 12 2012