OFFSET

1,2

COMMENTS

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.

The set of numbers with n nonprime substrings is finite. Proof: Evidently, each 1-digit binary number represents 1 nonprime substring. Hence, each (n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 2^n, such that all numbers > b have more than n nonprime substrings.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

FORMULA

a(n) >= A217102(n).

a(n) <= 2^n.

a(n) <= 2^min(6 + n/6, 20*floor((n+125)/126)).

a(n) <= 64*2^(n/6).

With m := floor(log_2(a(n))) + 1:

a(n+m+1) >= 2*a(n), if a(n) is even.

a(n+m) >= 2*a(n), if a(n) is odd.

EXAMPLE

(1) = 1, since 1 = 1_2 (binary) is the greatest number with 1 nonprime substring.

a(2) = 3 = 11_2 has 3 substrings in binary representation (1, 1 and 11), two of them are nonprime substrings (1 and 1), and 11_2 = 3 is the only prime substrings. 3 is the greatest number with 2 nonprime substrings.

a(8) = 29 = 11101_2 has 15 substrings in binary representation (0, 1, 1, 1, 1, 11, 11, 10, 01, 111, 110, 101, 1110, 1101, 11101), exactly 8 of them are nonprime substrings (0, 1, 1, 1, 1, 01, 110, 1110). There is no greater number with 8 nonprime substrings in binary representation.

a(14) = 54 = 110110_2 has 21 substrings in binary representation, only 7 of them are prime substrings (10, 10, 11, 11, 101, 1011, 1101), which implies that exactly 14 substrings must be nonprime. There is no greater number with 14 nonprime substrings in binary representation.

CROSSREFS

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Dec 20 2012

STATUS

approved