On Positive Integers Satisfying n = s(n) · τ(n): Anand Numbers

Abstract: We classify positive integers n satisfying n = s(n) · τ(n), where s(n) is the sum of decimal digits (base 10) and τ(n) is the number of positive divisors (OEIS A000005). We prove that exactly three positive integers satisfy this equation for n ≥ 1: {1, 80, 108}. We term such integers "Anand Numbers". The mathematical proof utilizes the classical upper bound τ(n) ≤ 2√n and digit-sum bounds, yielding an absolute theoretical upper bound of n ≤ 5184, which is verified computationally up to 10^6. The number 0 is explicitly excluded as τ(0) is undefined. The resulting triple consists entirely of tetradic digits {0,1,8} and exhibits interesting connections with Harshad numbers (A005349), refactorable numbers (A033950), and the third hyperfactorial H(3) = 108. No exact sequence matching this property was found in OEIS, MathWorld, or mathematical literature as of 25 October 2026. Future work will explore base-b generalizations and related functions. Deeper invariances of this triple are reserved for subsequent publication and the author's forthcoming book "Manujatva, Mrityu and Moksha (Yoga Darshan)". Keywords: Number Theory, Anand Numbers, Divisor Function, Digit Sum, Mathematical Research, Harshad Numbers, Refactorable Numbers.