Let φ be Euler’s function and fix an integer k≥0 . We show that for every initial value x1≥1 , the sequence of positive integers (xn)n≥1 defined by xn+1=φ(xn)+k for all n≥1 is eventually periodic. Similarly, for all initial values x1,x2≥1 , the sequence of positive integers (xn)n≥1 defined by xn+2=φ(xn+1)+φ(xn)+k for all n≥1 is eventually periodic, provided that k is even.
On iterates of shifted Euler's function
Paolo Leonetti
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2023-01-01
Abstract
Let φ be Euler’s function and fix an integer k≥0 . We show that for every initial value x1≥1 , the sequence of positive integers (xn)n≥1 defined by xn+1=φ(xn)+k for all n≥1 is eventually periodic. Similarly, for all initial values x1,x2≥1 , the sequence of positive integers (xn)n≥1 defined by xn+2=φ(xn+1)+φ(xn)+k for all n≥1 is eventually periodic, provided that k is even.
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