A112509 Maximum number of numbers represented by substrings of an n-bit number's binary representation.

1, 3, 5, 7, 10, 13, 17, 22, 27, 33, 40, 47, 55, 64, 73, 83, 94, 106, 118, 131, 145, 160, 176, 192, 209, 227, 246, 265, 285, 306, 328, 351, 375, 399, 424, 450, 477, 504, 532, 561, 591, 622, 654, 687, 720, 754, 789, 825, 862, 899, 937, 977, 1017, 1058, 1100, 1143

Offset: 1

Author

Rick L. Shepherd, Sep 09 2005

Keywords

• nonn
• base

Comments

Substrings must be contiguous, they are treated as stand-alone binary representations and the reversal of substrings is not permitted.

Examples

			To see why a(4)=7 (and A112510(4)=12 and A112511(4)=14), consider all numbers whose binary representations require exactly 4 bits: 1000, 1001, 1010, 1011, 1100, 1101, 1110 and 1111. For each of these binary representations in turn, we find the nonnegative integers represented by all of its contiguous substrings. We count these distinct integer values (putting the count in {}s):
1000: any 0, either 00, or 000 -> 0, 1 -> 1, 10 -> 2, 100 -> 4, 1000 -> 8 {5};
1001: either 0, or 00 -> 0, either 1, 01, or 001 -> 1, 10 -> 2, 100 -> 4, 1001 -> 9 {5};
(For brevity, binary substrings are shown below only if they produce values not shown yet.)
1010: 0, 1, 2, 101 -> 5, 1010 -> 10 {5};
1011: 0, 1, 2, 11 -> 3, 5, 1011 -> 11 {6};
1100: 0, 1, 2, 3, 4, 110 -> 6, 1100 -> 12 {7};
1101: 0, 1, 2, 3, 5, 6, 1101 -> 13 {7};
1110: 0, 1, 2, 3, 6, 111 -> 7, 1110 -> 14 {7};
1111: 1, 3, 7, 1111 -> 15 {4}.
Because the maximum number of distinct integer values {in brackets} is 7, a(4)=7. The smallest 4-bit number for which 7 distinct values are found is 12, so A112510(4)=12. The largest 4-bit number for which 7 are found is 14, so A112511(4)=14. (For n=4 the count is a(n)=7 also for all values (only one, 13, here) between A112510(n) and A112511(n). This is not the case in general.).
		

Links

• Martin Fuller, Table of n, a(n) for n = 1..80
• 2008/9 British Mathematical Olympiad Round 2, Problem 4, Jan 29 2009.

Crossrefs

Cf. A112510 (least n-bit number for which this maximum occurs), A112511 (greatest n-bit number for which this maximum occurs).

Cf. A078822, A122953, A156022, A156023, A156024, A156025. Equals A156022(n)+1 for n >= 2. [From Joseph Myers, Feb 01 2009]

Programs

Python

from itertools import product
def c(w):
    return len(set(w[i:j+1] for i in range(len(w)) if w[i] != "0" for j in range(i,len(w)))) + int("0" in w)
def a(n):
    return max(c("1"+"".join(b)) for b in product("01", repeat=n-1))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jan 13 2023

Extensions

a(21) to a(31) from Joseph Myers, Feb 01 2009

a(32) onwards from Martin Fuller, Jul 24 2025