ON COMPOSITE n DIVIDING '(n)(n) + 1
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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.
Abstract
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.
참고문헌 (3)
- Robin, G., "Estimation de la fonction de Tchebychef θ sur le k-leme nombre premies et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n", Acta Arih., 42, pp.367~389 (1983)
- Sandor, J., "Handbook of Number Theory II" (2004), Kluwer Academic Publishers
- Yong-Gao Chen, "on n | ϕ(n)d(n) + 2 and n | ϕ(n)σ(n) + 1, INTEGERS", Electronic J. Combinatorial Number Theory, 8, pp.A07~ (2008)