OFFSET
1,2
COMMENTS
Least integer k such that the number of iterations of Euler phi function needed to reach 1 starting at k (k is counted) is n.
a(n) is smallest number in the class k(n) which groups families of integers which take the same number of iterations of the totient function to evolve to 1. The maximum is 2*3^(n-1).
Shapiro shows that the smallest number is greater than 2^(n-1). Catlin shows that if a(n) is odd and composite, then its factors are among the a(k), k < n. For example a(12) = a(5) a(8). There is a conjecture that all terms of this sequence are odd. - T. D. Noe, Mar 08 2004
The indices of odd prime terms are given by n=A136040(k)+2 for k=1,2,3,.... - T. D. Noe, Dec 14 2007
Shapiro mentions on page 30 of his paper the conjecture that a(n) is prime for each n > 1, but a(13) is composite and so the conjecture fails. - Charles R Greathouse IV, Oct 28 2011
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.
R. K. Guy, Unsolved Problems in Number Theory, 2nd Ed. New York: Springer-Verlag, p. 97, 1994, Section B41.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1002
P. A. Catlin, Concerning the iterated phi-function, Amer Math. Monthly 77 (1970), pp. 60-61.
T. D. Noe, Primes in classes of the iterated totient function, JIS 11 (2008) 08.1.2
Harold Shapiro, An arithmetic function arising from the phi function, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.
FORMULA
EXAMPLE
a(3) = 3 because trajectory={3,2,1}. n=1: a(1)=1 because trajectory={1}
MATHEMATICA
f[n_] := Length[ NestWhileList[ EulerPhi, n, Unequal, 2]] - 1; a = Table[0, {30}]; Do[b = f[n]; If[a[[b]] == 0, a[[b]] = n; Print[n, " = ", b]], {n, 1, 22500000}] (* Robert G. Wilson v *)
PROG
(Haskell)
a007755 = (+ 1) . fromJust . (`elemIndex` a003434_list) . (subtract 1)
-- Reinhard Zumkeller, Feb 08 2013, Jul 03 2011
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Pepijn van Erp [ vanerp(AT)sci.kun.nl ]
EXTENSIONS
More terms from David W. Wilson, May 15 1997
Additional comments from James S. Cronen (cronej(AT)rpi.edu)
STATUS
approved