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ON COMPOSITE n DIVIDING '(n)(n) + 1
V.Siva Rama Prasad,C. Goverdhan,Hussain Abdulkader Al-Aidroos 장전수학회 2013 Advanced Studies in Contemporary Mathematics Vol.23 No.1
Let ' denote the Euler’'s totient function. For any positive integer n, if (n) denotes the sum of all its positive divisors, and Rk = {n : '(n)(n) + 1 = kn} for k = 1, 2, 3, ... and R =1[k=1Rk it has been proved in [3] that every composite integer n 2 R if it exists, has at least three distinct odd prime factors. In this note we improve this result for n 2 Rk in the cases (a) 5 | n and k 336 and (b) 5 - n and k 672. Also we give lower bounds for n in both the cases.
On Lehmer’s totient problem and its Unitary analogue
V. Siva Rama Prasad,C. Goverdhan,Hussain Abdulkader Al-Aidroos 장전수학회 2010 Proceedings of the Jangjeon mathematical society Vol.13 No.2
Let Ø be the Euler’s totient function and Ø* be its unitary analogue. For any integer M ≥ 1, let SM = {n ∈ N :MØ(n) = n − 1} and S*M = {n ∈ N : MØ*(n) = n − 1}. If n ∈ S*M with M ≥ 4, we obtain lower bounds for n and ω(n) ( the number of distinct prime factors of n) which significantly improve those given in [1] and [5] for n in SM and S*M respectively.