The reciprocal sum of primitive nondeficient numbers

Abstract

We investigate the reciprocal sum of primitive nondeficient numbers, or pnds. In 1934, Erdős showed that the reciprocal sum of pnds converges, which he used to prove that the abundant numbers have a natural density. We show the reciprocal sum of pnds is between 0.348 and 0.380.

The Erdős conjecture for primitive sets

“…Thus, the sum of densities for each S a is dominated by g(q), that is, gives the density of nondeficient numbers recently shown in [12] to lie in the tight interval (0.2476171, 0.2476475). In [14], an analog of Proposition 2.1 is a key ingredient for sharp bounds on the reciprocal sum of the primitive nondeficient numbers.…”

Untitled

“…Reciprocal sums reflect the distribution of number sets. Convergent sums have been bounded for consecutive primes differing by 2 [1], amicable pairs [8] and primitive nondeficient numbers [7] that are consistent with the order estimates for the cardinality of the sets less than a given integer [2].…”

Must a primitive non-deficient number have a component not much larger than its radical?

“…[2] Prior work on these sets have looked at their density. See especially [1] as well as [7] and [11]. In fact, the arguments used in this paper can be thought of as variations of the techniques used in [1].…”