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.

0, 1, 3, 5, 8, 10, 13, 16, 19, 22, 26, 29, 33, 36, 40, 44, 49, 52, 56, 60, 64, 68, 73, 77, 82, 86, 90, 94, 99, 103, 108, 113, 118, 123, 128, 132, 137, 141, 146, 151, 157, 161, 166, 171, 176, 181, 187, 192, 197, 202, 208, 213, 219, 223, 229, 234, 239, 244, 250, 255

Offset: 0

Author

Labos Elemer, Apr 13 2001

Keywords

• nonn

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Hartosh Singh Bal and Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019. See function S(n), p. 2.
• Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
• Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
• Harold Shapiro, An arithmetic function arising from Phi-function, American Math. Monthly, Vol. 50, No. 1 (1943), pp. 18-30.

Crossrefs

Cf. A003434, A049108.

Programs

Mathematica

f[1] = 0; f[n_] := f[n] = f[EulerPhi[n]] + 1; Accumulate[Array[f, 100]] (* Amiram Eldar, Nov 27 2024 *)

Formula

a(n) = Sum_{j=1..n} A003434(j).