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A092693 Sum of iterated phi(n).

%I A092693 #41 Nov 15 2021 01:26:48
%S A092693 0,1,3,3,7,3,9,7,9,7,17,7,19,9,15,15,31,9,27,15,19,17,39,15,35,19,27,
%T A092693 19,47,15,45,31,35,31,39,19,55,27,39,31,71,19,61,35,39,39,85,31,61,35,
%U A092693 63,39,91,27,71,39,55,47,105,31,91,45,55,63,79,35,101,63,79,39,109,39,111
%N A092693 Sum of iterated phi(n).
%C A092693 Iannucci, Moujie and Cohen examine perfect totient numbers: n such that a(n) = n.
%H A092693 T. D. Noe, <a href="/A092693/b092693.txt">Table of n, a(n) for n = 1..10000</a>
%H A092693 C. Defant, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Defant/defant5.html">On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions</a>, J. Int. Seq. 18 (2015) # 15.2.1
%H A092693 P. Erdos and M. V. Subbarao, <a href="http://www.math.ualberta.ca/~subbarao/documents/Subbarao1.pdf">On the iterates of some arithmetic functions</a>, The theory of arithmetic functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich. 1971), Lecture Notes in Math., 251 , pp. 119-125, Springer, Berlin, 1972. [<a href="http://www.renyi.hu/~p_erdos/1972-17.pdf">alternate link</a>]
%H A092693 Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Cohen2/cohen50.html">On perfect totient numbers</a>, J. Integer Sequences, 6 (2003), #03.4.5.
%F A092693 a(1) = 0, a(n) = phi(n) + a(phi(n))
%F A092693 a(n) = A053478(n) - n. - _Vladeta Jovovic_, Jul 02 2004
%F A092693 Erdős & Subbarao prove that a(n) ~ phi(n) for almost all n. In particular, a(n) < n for almost all n. The proportion of numbers up to N for which a(n) > n is at most 1/log log log log N. - _Charles R Greathouse IV_, Mar 22 2012
%e A092693 a(100) = 71 because the iterations of phi (40, 16, 8, 4, 2, 1) sum to 71.
%t A092693 nMax=100; a=Table[0, {nMax}]; Do[e=EulerPhi[n]; a[[n]]=e+a[[e]], {n, 2, nMax}]; a (* _T. D. Noe_ *)
%t A092693 Table[Plus @@ FixedPointList[EulerPhi, n] - (n + 1), {n, 72}] (* _Alonso del Arte_, Jan 29 2007 *)
%o A092693 (Haskell)
%o A092693 a092693 1 = 0
%o A092693 a092693 n = (+ 1) $ sum $ takeWhile (/= 1) $ iterate a000010 $ a000010 n
%o A092693 -- _Reinhard Zumkeller_, Oct 27 2011
%o A092693 (PARI) a(n)=my(k);while(n>1,k+=n=eulerphi(n));k \\ _Charles R Greathouse IV_, Mar 22 2012
%o A092693 (Python)
%o A092693 from sympy import totient
%o A092693 from math import prod
%o A092693 def f(n):
%o A092693     m = n
%o A092693     while m > 1:
%o A092693         m = totient(m)
%o A092693         yield m
%o A092693 def A092693(n): return sum(f(n)) # _Chai Wah Wu_, Nov 14 2021
%Y A092693 Cf. A003434 (iterations of phi(n) needed to reach 1), A092694 (iterated phi product).
%Y A092693 Cf. A082897 and A091847 (perfect totient numbers).
%K A092693 nonn
%O A092693 1,3
%A A092693 _T. D. Noe_, Mar 04 2004