An Arithmetic Function Arising from the Dedekind $\psi$ Function

zbMATH Open1419.11007arXiv1501.00971MaRDI QIDQ2961012

Publication date: 17 February 2017

Abstract: We define

o v e r l i n e p s i

to be the multiplicative arithemtic function that satisfies [overline{psi}(p^{alpha})=�egin{cases} p^{alpha-1}(p+1), & mbox{if } p eq 2; \ p^{alpha-1}, & mbox{if } p=2 end{cases}] for all primes

p

and positive integers

a l p h a

. Let

l a m b d a (n)

be the number of iterations of the function

o v e r l i n e p s i

needed for

n

to reach

2

. It follows from a theorem due to White that

l a m b d a

is additive. Following Shapiro's work on the iterated

v a r p h i

function, we determine bounds for

l a m b d a

. We also use the function

l a m b d a

to partition the set of positive integers into three sets

S_{1}, S_{2}, S_{3}

and determine some properties of these sets.

Full work available at URL: https://arxiv.org/abs/1501.00971

zbMATH Keywords

additive function Dedekind function iterated function

Mathematics Subject Classification ID

Arithmetic functions; related numbers; inversion formulas (11A25)