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a(n) = sum of lengths of the iteration sequences of Euler totient function from 1 to n.
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%I #22 Mar 23 2024 12:56:48

%S 1,3,6,9,13,16,20,24,28,32,37,41,46,50,55,60,66,70,75,80,85,90,96,101,

%T 107,112,117,122,128,133,139,145,151,157,163,168,174,179,185,191,198,

%U 203,209,215,221,227,234,240,246,252,259,265,272,277,284,290,296,302

%N a(n) = sum of lengths of the iteration sequences of Euler totient function from 1 to n.

%C Partial sums of A049108. - _Joerg Arndt_, Jan 06 2015

%H 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 Paul Erdos, 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 Harold Shapiro, <a href="http://www.jstor.org/stable/2303988">An arithmetic function arising from the phi function</a>, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.

%F a(n) = sum( j=1..n, A049108(j) ).

%e Iteration sequences of Phi applied to 1, 2, 3, 4, 5, 6 give lengths 1, 2, 3, 3, 4, 3 with partial sums as follows:1, 3, 5, 9, 13, 16 resulting in first...6th terms here.

%t Accumulate[Table[Length[NestWhileList[EulerPhi,n,#!=1&]],{n,60}]] (* _Harvey P. Dale_, Mar 23 2024 *)

%o (PARI) a049108(n)=my(t=1); while(n>1, t++; n=eulerphi(n)); t;

%o vector(80, n, sum(j=1, n, a049108(j))) \\ _Michel Marcus_, Jan 06 2015

%Y Cf. A049108, A003434.

%K nonn

%O 1,2

%A _Labos Elemer_, Apr 13 2001