A generalization of the Erdos-Serpinski conjecture

Subject Areas : Statistics

Hamid Torabi ¹ , AmirAli Fatehizadeh ²

1 - Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159- 91775, Mashhad, Iran.
2 - Department of Mathematics, Quchan University of Technology, Quchan, Iran.

Received: 2019-11-01 Accepted : 2020-11-03 Published : 2021-02-19

Keywords: اعداد کا-تام, تابع ضربی, تابع مجموع مقسوم علیه های یک عدد, اعداد تام,

Abstract :

Abstract: Suppose that σ(n) is the sum of the divisors of n. This paper focuses on the Erdos-Serpinsky conjecture, which expresses the set of solutions of equation σ(n+1)=σ(n) is infinite. In present paper, we review some research on solutions of equations involving σ. As a generalization of equation σ(n+1)=σ(n), we investigate solutions of equation σ(n+1)=σ(n) under various conditions. For example, by using the representation of perfect numbers, we show that for a prime number n, n is a solution of equation σ(n+1)=2σ(n), if and only if n is equals to 5. Consequently, we conclude that for a prime number n≠5, n is a solution of equation σ(n+1)=kσ(n) if and only if n+1 is a k-perfect number. Also, we show that the only solution of equation σ(n+1)=2^r σ(n) which is presented as n=p ,n+1=2q_1 q_2…q_s ,where s≤r and q_1 ، q_2،...، q_s and p are odd and prime numbers, is (n,r)=(5,1).

References:

[1] L. Benito, Solutions of the problem of Erdos-Sierpinski:  arXiv: 0707.2190v1.

 

[2] D. M. Burton, Elementary number theory, Mc Grow Hil, 2007.

 

[3] K. Ford, The number of solutions of (x) = m, Annals of Mathematics, 150 (1999) 1-29.

 

[4] K. Ford, Florian Luca, and Carl Pomerance, Common values of the arithmetic functions φ and , Bull. Lond. Math. Soc. 42 (2010), 478–488.

 

[5] R. Guy and D. Shanks, A constructed solution of , Fibonacci Quart., 12 (1974) 299-299.

 

[6] P. Haukkanen, Some computational results concerning the divisor functions d(n) and σ (n), Math. Student 62 (1993), 166–168.

 

[7] J.L. Hunsucker, J. Nebb, and R. E. Stearns, Computational results concerning some equations involving σ(n), Math. Student 41 (1973), 285-28.

 

[8] D. Kim, U. Sarp, and S. Ikikardes, Certain combinatoric convolution sums arising from Bernoulli and Euler polynomials, Miskolic Mathematical Notes, 20 (2019) 311-330.

 

[9] W. E. Mientka and R. L. Vogt, Computational results relating to problems concerning σ(n), Mat. Vesnik 7 (1970), 35–36.

 

[10] P. Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. 60 (2011) 199–214.

 

[11] P. Pollack, Some arithmetic properties of the sum of proper divisors and the sum of prime divisors, Illinois J. Math. 58 (2014) 125–147.

 

[12] P. Pollack and C. Pomerance, Prime-perfect numbers, Integers 12 (2012) 1417– 1437.

 

[13] P. Pollack and C. Pomerance, Some problems of Erdos on the sum-of-divisors function, Transactions of the American Mathematical Society, 7 (2016), 1-26. 

 

[14] P. Pollack, M. Rassias and C. Pomerance, Remarks on fibers of the sum-of-divisors function, in Analytic number theory, Springer. Cham. Switzerland, (2015) 305–320.

 

[15] P. Pollack and L. Thompson, Arithmetic functions at consecutive shifted primes, International Journal of Number Theory, 11 (2015),1477-1498.

 

[16] C. Pomerace and H. Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.

 

[17] A. Weingartner, On the Solutions of , Journal of Integer Sequences, 14 (2011) 1-7.