Unitary harmonic numbers

Unitary untouchable numbers

In 1973, Erdős proved that a positive proportion of numbers are untouchable; that is, not of the form s(n), where s(n) := σ(n)−n is the sum of the proper divisors of n. We investigate the analogous question where σ is replaced with similar divisor functions, such as the unitary sum-of-divisors function σ∗ (which sums those divisors d of n co-prime to n/d). We use the slightly modified version o...

Some summation formulas involving harmonic numbers and generalized harmonic numbers

Summation formulae involving harmonic numbers

Several summation formulae for finite and infinite series involving the classical harmonic numbers are presented. The classical harmonic numbers are defined by

Odd harmonic numbers exceed 1024

A number n > 1 is harmonic if σ(n) | nτ(n), where τ(n) and σ(n) are the number of positive divisors of n and their sum, respectively. It is known that there are no odd harmonic numbers up to 1015. We show here that, for any odd number n > 106, τ(n) ≤ n1/3. It follows readily that if n is odd and harmonic, then n > p3a/2 for any prime power divisor pa of n, and we have used this in showing that ...

On Differentiation and Harmonic Numbers

= (−1)(1 + n+m), where Hn := 1 + 1 2 + · · · + 1 n . For both identities we have the condition n ≥ m ≥ 1. In [M], these identities were crucial in proving several Beukers like supercongruences that had been observed numerically by Fernando Rodriguez-Villegas [FRV]. In [M], these identities were broken up into smaller pieces, and each part was evaluated using Wilf-Zeilberger [PWZ] theory. Althou...