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A003434
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Number of iterations of phi(x) at n needed to reach 1.
(Formerly M0244)
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39
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0, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 3, 4, 4, 5, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 4, 5, 4, 5, 5, 6, 4, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 6, 4, 6, 5, 5, 5, 6, 5, 6, 5, 5, 6, 6, 5, 6, 6, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 5, 6, 7, 5, 7, 5, 6, 6, 7, 5, 6, 6, 6, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 6, 6
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OFFSET
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1,3
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COMMENTS
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Each number n>1 occurs for the first time at the position A007755(n+1) and for the last time at 2*3^(n-1). - Ivan Neretin, Mar 24 2015
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. V. Subbarao, On a function connected with phi(n), J. Madras Univ. B. 27 (1957), pp. 327-333.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Hartosh Singh Bal, Gaurav Bhatnagar, Prime number conjectures from the Shapiro class structure, arXiv:1903.09619 [math.NT], 2019.
C. Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015) # 15.2.1.
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]
I. Niven, The iteration of certain arithmetic functions, Canad. J. Math. 2 (1950), pp. 406-408.
H. N. Shapiro, On the iterates of a certain class of arithmetic functions, Comm. Pure Appl. Math. 3 (1950), pp. 259-272.
S. Sivasankaranarayana Pillai, On a function connected with phi(n), Bull. Amer. Math. Soc., 35:6 (1929), pp. 837-841.
S. Sivasankaranarayana Pillai, On a function connected with phi(n), Bull. Amer. Math. Soc., 35.6 (1929), 837-841. (Annotated scanned copy)
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FORMULA
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a(n) = A049108(n) - 1.
By the definition of a(n) we have for n >= 2 the recursion a(n) = a(phi(n)) + 1. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001
Pillai proved that log(n/2)/log(3) + 1 <= a(n) <= log(n)/log(2) + 1. - Charles R Greathouse IV, Mar 22, 2012
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EXAMPLE
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If n=164 the trajectory is {164,80,32,16,8,4,2,1}. Its length is 8, thus a(164)=7.
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MAPLE
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A003434 := proc(n)
local a, e;
e := n ;
a :=0 ;
while e > 1 do
a := a+1 ;
e := numtheory[phi](e) ;
end do:
a;
end proc:
seq(A003434(n), n=1..40) ; # R. J. Mathar, Jan 09 2017
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MATHEMATICA
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f[n_] := Length@ NestWhileList[ EulerPhi, n, # != 1 &] - 1; Array[f, 105] (* Robert G. Wilson v, Feb 07 2012 *)
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PROG
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(PARI) A003434(n)=for(k=0, n, n>1 || return(k); n=eulerphi(n)) /* Works because the loop limits are evaluated only once. Using while(...) takes 50% more time. */ \\ M. F. Hasler, Jul 01 2009
(Haskell)
a003434 n = fst $ until ((== 1) . snd)
(\(i, x) -> (i + 1, a000010 x)) (0, n)
-- Reinhard Zumkeller, Feb 08 2013, Jul 03 2011
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CROSSREFS
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Cf. A000010, A007755.
Sequence in context: A235613 A322418 A019569 * A330808 A097849 A100678
Adjacent sequences: A003431 A003432 A003433 * A003435 A003436 A003437
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KEYWORD
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nonn,easy,nice,look
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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