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The greatest prime factor and recurrent sequences. (English) Zbl 1221.11032

The authors consider sequences \(p_j = \text{gpf}(a_1p_j - 1+a_2p_{j - 2}+\cdots +a_dp_{j - d}+a_0)\), where for any integer \(x\geq 2\), gpf(\(x\)) denotes the greatest prime factor of \(x\). In the simple case of the ‘GPF-Fibonacci’ sequences corresponding to \(d = 2, a_0 = 0\), and \(a_1 = a_2 = 1\), they find that regardless of the initial conditions \(p_0\) and \(p_1\), all such sequences ultimately enter the cycle 7, 3, 5, 2. A computational exploration of the ‘GPF-Tribonacci’ analogue \(d = 3, a_0 = 0\), and \(a_1 = a_2 = a_3 = 1\) reveals four cycles of lengths, listed in the decreasing order of frequencies, 100, 212, 28 and 6, with the two larger cycles collecting more than 98% of the sequences as defined by the initial conditions \(p_0, p_1\), and \(p_2\). The paper concludes with a general ultimate periodicity conjecture and discusses its plausibility.

MSC:

11B37 Recurrences
11A41 Primes
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