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A060606 The n-th term is the sum of lengths of iteration chains to get fixed points (=1) for the Euler totient function from 1 to n.

%I A060606 #26 Nov 27 2024 07:27:10
%S A060606 0,1,3,5,8,10,13,16,19,22,26,29,33,36,40,44,49,52,56,60,64,68,73,77,
%T A060606 82,86,90,94,99,103,108,113,118,123,128,132,137,141,146,151,157,161,
%U A060606 166,171,176,181,187,192,197,202,208,213,219,223,229,234,239,244,250,255
%N A060606 The n-th term is the sum of lengths of iteration chains to get fixed points (=1) for the Euler totient function from 1 to n.
%H A060606 Amiram Eldar, <a href="/A060606/b060606.txt">Table of n, a(n) for n = 0..10000</a>
%H A060606 Hartosh Singh Bal and Gaurav Bhatnagar, <a href="https://arxiv.org/abs/1903.09619">Prime number conjectures from the Shapiro class structure</a>, arXiv:1903.09619 [math.NT], 2019. See function S(n), p. 2.
%H A060606 Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="http://math.dartmouth.edu/~carlp/iterate.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
%H A060606 Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
%H A060606 Harold Shapiro, <a href="https://www.jstor.org/stable/2303988">An arithmetic function arising from Phi-function</a>, American Math. Monthly, Vol. 50, No. 1 (1943), pp. 18-30.
%F A060606 a(n) = Sum_{j=1..n} A003434(j).
%e A060606 Iteration sequences of Phi applied to 1,2,3,4,5,6 give lengths 0,1,2,2,3,2 with partial sums as follows:0,1,3,5,8,10 resulting in the first six terms of this sequence. It differs by n from the analogous sums applied to A049108 sequence.
%t A060606 f[1] = 0; f[n_] := f[n] = f[EulerPhi[n]] + 1; Accumulate[Array[f, 100]] (* _Amiram Eldar_, Nov 27 2024 *)
%Y A060606 Cf. A003434, A049108.
%K A060606 nonn
%O A060606 0,3
%A A060606 _Labos Elemer_, Apr 13 2001