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

0,3

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.

Formula

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

Example

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.

Mathematica

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

Crossrefs

Cf. A003434, A049108.

Sequence in context: A245314 A186150 A377206 * A350234 A050504 A330038

Adjacent sequences: A060603 A060604 A060605 * A060607 A060608 A060609

Keyword

nonn

Author

Labos Elemer, Apr 13 2001

Status

approved