Generalized Perfect Numbers Connected with Arithmetic Functions

On new arithmetic function relative to a fixed positive integer. Part 1

“…Many mathematicians have been studied the perfect numbers and their generalizations with the help of various arithmetic functions (see e.g., [7,8,11]). In [9,10], some arithmetic functions are used in characterizing generalized Mersenne primes.…”

Connected components of complex divisor functions

“…Many early theorems in number theory spawned from attempts to understand perfect numbers. Although few modern mathematicians continue to attribute the same theological or mystical significance to perfect numbers that ancient people once did, these numbers remain a substantial inspiration for research in elementary number theory [2,3,5,11,[14][15][16]18]. Around 100 A.D., Nicomachus stated that perfect numbers represent a type of "harmony" between "deficient" and "abundant" numbers.…”

“…If s is the unique real number such that s ≥ T c and w( c (s)) = w(z k ) (in other words, s is the unique real number such that s ≥ T c and z k ∈ (w( c (s)), c (s)]), then (11) c q…”