|
| |
|
|
A003434
|
|
Number of iterations of phi(x) at n needed to reach 1.
(Formerly M0244)
|
|
28
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
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
|
|
|
REFERENCES
|
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.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n = 1..10000
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 Erdos, 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)
|
|
|
FORMULA
|
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
|
|
|
EXAMPLE
|
If n=164 the trajectory is {164,80,32,16,8,4,2,1}. Its length is 8, thus a(164)=7.
|
|
|
MAPLE
|
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
|
|
|
MATHEMATICA
|
f[n_] := Length@ NestWhileList[ EulerPhi, n, # != 1 &] - 1; Array[f, 105] (* Robert G. Wilson v, Feb 07 2012 *)
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
Cf. A000010, A007755.
Sequence in context: A202472 A235613 A019569 * A097849 A100678 A026834
Adjacent sequences: A003431 A003432 A003433 * A003435 A003436 A003437
|
|
|
KEYWORD
|
nonn,easy,nice,look
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
STATUS
|
approved
|
| |
|
|
