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ON THE ALTERNATING SUM-OF-DIVISORS
Keywords:
alternating sum-of-divisors, totient function.DOI:
https://doi.org/10.17654/0972555524006Abstract
We show that the alternating sum-of-divisors
$$
\chi(N)=d_1-d_2+d_3-\cdots+(-1)^{m-1} d_m,
$$
where $N=d_1>d_2>d_3>\cdots>d_m=1$, even if not a multiplicative function, has good factorization properties for a special class of integers $N$ which we call "of a super-increasing type" with factorization $N=p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ with $p_1<p_2<\cdots<p_k$ and $p_{i+1}>\prod_{j \leq i} p_j^{\alpha_j}$ for $i=1,2, \ldots, k-1$. As a consequence, for this class of integers, the open problem $\chi(N) \geq \varphi(N)$, where $\varphi$ represents the totient function, is solved affirmatively.
Received: November 20, 2023
Revised: December 1, 2023
Accepted: December 26, 2023
References
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford University Press, 1960.
L. Toth, A survey of the alternating sum-of-divisors function, Acta Univ. Sapientiae Math. 5(1) (2013), 93-107.
K. Atanassov, A remark on an arithmetic function. Part 2, Notes on Number Theory and Discrete Mathematics 15(3) (2009), 21-22.
D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, 2007.
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