Self-Healing Numbers: A New Class of Integers with Positional Divisibility Properties

1. Introduction

The study of integers with special divisibility properties has been a recurring theme in number theory. Classical examples include the divisibility tests for small primes, Harshad numbers (integers divisible by the sum of their digits), and self-divisive numbers (integers divisible by each of their individual digits).

We introduce a variant based on positional digit removal and divisibility.

Definition 1.1.  Let n be a positive integer with decimal representation d₁d₂ . . . d_k where d₁ ̸= 0. For each i∈ {1, 2, . . . , k}, let n⁽ⁱ⁾ denote the (k − 1)-digit integer obtained by removing the digit d_i from position i. We say that n is a Self-Healing Number (SHN) if i | n⁽ⁱ⁾ for all i∈ {1, 2, . . . , k}.

Example 1.2.The integer n = 20 is self-healing:

n(1) = 0,	1 | 0	(1)
n(2) = 2,	2 | 2	(2)

Example 1.3.The integer n = 111 is not self-healing:

n(1) = 11,	1 | 11	(3)
n(2) = 11,	2 ∤ 11	(4)

Since 11 ≢ 0 (mod 2), the number 111 fails the condition for i = 2.

2. Basic Properties and Congruence Relations

Theorem 2.1.  Every single-digit positive integer is self-healing.

Proof. For any n ∈ {1, 2, . . . , 9}, removing the digit at position 1 yields n⁽¹⁾ = 0. Since 1 | 0, the divisibility condition is satisfied. □

Theorem 2.2.  A two-digit integer n = 10a + b with a∈ {1, 2, . . . , 9} and b∈ {0, 1, . . . , 9} is self-healing if and only if a is even.

Proof. For n = 10a + b, we have:

n⁽¹⁾ = b

(5)

n⁽²⁾ = a

(6)

The condition 1 | n⁽¹⁾ = b is always satisfied. The condition 2 | n⁽²⁾ = a requires a to be even. Therefore, n is self-healing if and only if a ∈ {2, 4, 6, 8}. □

Corollary 2.3.  There are exactly 40 two-digit self-healing numbers, namely:

{20, 21, . . . , 29, 40, 41, . . . , 49, 60, 61, . . . , 69, 80, 81, . . . , 89}

Proposition 2.4. Let n = d₁d₂ . . . d_k be a k-digit integer. The integer n⁽ⁱ⁾ can be expressed as:

i−1

n(i) = dj 10^k−1−j +

j=1

k

L

j=i+1 dj 10^k−j

The divisibility condition i | n⁽ⁱ⁾ is equivalent to the congruence relation n⁽ⁱ⁾ ≡ 0 (mod i). For i > 1, this can be simplified. For example, for a 3-digit integer n = 100d₁ + 10d₂ + d₃, the conditions are:

• 1 | 10d₂ + d₃ (always true)
• 2 | 100d1 + d₃ =⇒ d₃ ≡ 0 (mod 2)
• 3 | 100d1 + 10d₂ =⇒ d₁ + d₂ ≡ 0 (mod 3)

Proof. The formula for n⁽ⁱ⁾ is a direct consequence of the positional representation of integers. The simplification of the congruences uses standard divisibility rules and the fact that 10 ≡ 1 (mod 3) and 100 ≡ 1 (mod 3). For i = 2, 100 ≡ 0 (mod 2), so 2 | 100d₁ + d₃ simplifies to 2 | d₃. For i = 3,

100 ≡ 1 (mod 3) and 10 ≡ 1 (mod 3), so 3 | 100d₁ + 10d₂ simplifies to 3 | d₁ + d₂. □

3. Computational Results

We have performed exhaustive computational enumeration of self-healing numbers through seven digits. Each integer in the specified ranges was tested algorithmically according to Definition 1.1.

Table 1. Counts of Self-Healing Numbers by digit length.

Digit length k	Count a(k)	Cumulative
1	9	9
2	40	49
3	150	199
4	858	1,057
5	4,146	5,203
6	19,908	25,111
7	95,526	120,637

Table 1. Counts of Self-Healing Numbers by digit length.

Digit length k	Count a(k)	Cumulative
1	9	9
2	40	49
3	150	199
4	858	1,057
5	4,146	5,203
6	19,908	25,111
7	95,526	120,637

The sequence of self-healing numbers begins:

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, . . .

The counting function a(k) exhibits substantial growth.

4. Structural Properties

Theorem 4.1. Every self-healing number with two or more digits must have an even last digit.

Proof. Let n be an SHN with k ≥ 2 digits. By definition, n must satisfy the divisibility condition for i = 2, which means 2 | n⁽²⁾. The number n⁽²⁾ is obtained by removing the second digit of n. The parity (even or odd) of an integer is determined solely by its last digit. Since n⁽²⁾ has the same last digit as n, and n⁽²⁾ must be even, the last digit of n must also be even. □

Theorem 4.2. The self-healing property is not hereditary. That is, if n is self-healing, then prefixes of n need not be self-healing.

Proof. We exhibit a counterexample. Consider n = 152, and we verify that it is an SHN:

n(1) = 52,	1 | 52	(7)
n(2) = 12,	2 | 12	(8)
n(3) = 15,	3 | 15	(9)

Since all conditions are met, 152 is a self-healing number.

However, the two-digit prefix 15 is not self-healing:

15(1) = 5,	1 | 5	(10)
15(2) = 1,	2 ∤ 1	(11)

Since 1 ≢ 0 (mod 2), the number 15 is not self-healing. This completes the proof. □

5. Growth Analysis

The data in Table 1 reveals significant growth in the counting function. To analyze this growth, we examine the ratios a(k + 1)/a(k):

Table 2. Growth ratios for consecutive digit lengths.

k	a(k + 1)/a(k)
1	4.44
2	3.75
3	5.72
4	4.83
5	4.80
6	4.80

Table 2. Growth ratios for consecutive digit lengths.

k	a(k + 1)/a(k)
1	4.44
2	3.75
3	5.72
4	4.83
5	4.80
6	4.80

The growth ratios suggest exponential behavior, with ratios stabilizing around 4.8 for lengths 4-6.

Let ρ(k) = a(k)/(9 × 10^k−1) denote the density of self-healing numbers among k-digit integers.

We observe:

ρ(1) = 1.000, ρ(2) = 0.444, ρ(3) = 0.167, ρ(4) = 0.095

While the density decreases with k, it decreases slowly enough that the absolute count continues to grow substantially.

Conjecture 5.1. The sequence of self-healing numbers is infinite.

The sustained growth through seven digits and the regular pattern in growth ratios provide empirical support for this conjecture, though a proof remains elusive. A naive probabilistic model would suggest that the number of k-digit SHNs, a(k), is proportional to 10^k−1 , leading to a growth k+1 . The discrepancy between this simple model and our empirical data (which shows

a ratio stabilizing around 4.8) suggests that the divisibility conditions are not independent and require a more sophisticated analysis.

6. Algorithmic Considerations

The verification of whether an n-digit integer is self-healing requires checking n divisibility condi- tions. For each position i, we must:

• 1. Remove the digit at position i
• Convert the resulting (n − 1)-digit string to an integer
• 3. Check divisibility by i

The time complexity for testing a single n-digit number is O(n²), accounting for the digit manipulations and divisibility tests. Exhaustive enumeration of all k-digit self-healing numbers requires O(k² · 10^k) operations.

Theorem 4.1 provides a crucial computational optimization: we need only examine integers

with even terminal digits, reducing the search space by a factor of 2.

7. Related Work

Self-healing numbers are related to several existing classes of integers:

• Harshad numbers: integers divisible by the sum of their digits
• Self-divisive numbers: integers divisible by each of their nonzero digits
• Polydivisible numbers: k-digit numbers where the first j digits form a number divisible by j for j = 1, 2, . . . , k

The key distinction is that self-healing numbers involve removal of digits rather than manipu- lation or concatenation, leading to different mathematical behavior.

8. Open Problems

Several questions remain open:

• Prove or disprove Conjecture 5.
• Determine the asymptotic growth rate of a(k) as k → ∞.
• Characterize the distribution of self-healing numbers within each digit length.
• Investigate analogous sequences in bases other than 10.
• Find connections to deeper number-theoretic structures.

9. Conclusion

We have introduced self-healing numbers as a new class of integers with interesting positional divisibility properties. Complete enumeration through seven digits reveals substantial growth and several structural theorems. The non-hereditary nature and terminal digit constraints distinguish these numbers from related sequences in the literature.

The substantial growth observed computationally, combined with the structural properties we have established, suggests that self-healing numbers form a rich class worthy of further mathemat- ical investigation.