Vincenzo Manto

Breakable numbers

Created: 4/15/2026 | Author: Vincenzo Manto , Apr 06 2026

Smallest number k > n such that the binary Hamming distance between n and k is equal to the number of runs in the binary representation of n, and k has more runs than n.

Formal Definition

The smallest number k > n such that the binary Hamming distance between n and k is equal to the number of runs in the binary representation of n, and k has strictly more runs than n.

a(n) = min { k > n : dH(n, k) = R(n) and R(k) > R(n) }

Structural Analysis & Motivation

The designation "Breakable Numbers" stems from an intuition regarding the structural integrity of binary blocks. In computer science and combinatorics, a run defines a consecutive sequence of identical bits (either all 1s or all 0s).

Analytical Examples

  • For n = 1: Binary representation is 1, which has exactly R(1) = 1 run. We look for the smallest k > 1 such that changing 1 bit increases the run count.
    • Testing k = 5 (binary 101): The Hamming distance dH(1, 5) = 1 (since 001 to 101 requires flipping only the most significant bit). The number of runs in 101 is 3. Since 3 > 1, a(1) = 5.
  • For n = 2: Binary representation is 10, which has R(2) = 2 runs. We must flip exactly 2 bits to find a k with more than 2 runs.
    • Testing k = 11 (binary 1011): The Hamming distance dH(2, 11) = 2 (from 0010 to 1011). The number of runs in 1011 is 3. Since 3 > 2, a(2) = 11.

Sequence Chart

Graph of A394905

Data

5,11,11,13,22,10,23,11,21,45,18,20,20,22,47,19,22,43,20,45,90,37,26,27,40,41,41,44,37,46,95,35,38,45,36,43,86,41,41,53,85,181,82,69,74,53,53,51,54,74,52,69,82,83,58,59,73,75,75,92,77,94,191,67,70

Formula

a(n) = min { k > n : d_H(n, k) = R(n) and R(k) > R(n) }.

Cross-References

See also OEIS entries: Cf. A044813, A005811.