A007755 Smallest number m such that the trajectory of m under iteration of Euler's totient function phi(n) [A000010] contains exactly n distinct numbers, including m and the fixed point.

1, 2, 3, 5, 11, 17, 41, 83, 137, 257, 641, 1097, 2329, 4369, 10537, 17477, 35209, 65537, 140417, 281929, 557057, 1114129, 2384897, 4227137, 8978569, 16843009, 35946497, 71304257, 143163649, 286331153, 541073537, 1086374209, 2281701377, 4295098369

Offset: 1

Author

Pepijn van Erp [ vanerp(AT)sci.kun.nl ]

Keywords

• nonn
• nice

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

Examples

			a(3) = 3 because trajectory={3,2,1}. n=1: a(1)=1 because trajectory={1}
		

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, Computing Numbers in Section I of the Totient Iteration
• 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.

Crossrefs

Cf. A000010, A003434, A049108, A092873 (prime factors of a(n)), A060611, A098196, A227946.

A060611 has the same initial terms but is a different sequence.

Programs

Haskell

a007755 = (+ 1) . fromJust . (`elemIndex` a003434_list) . (subtract 1)
-- Reinhard Zumkeller, Feb 08 2013, Jul 03 2011

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 *)

Formula

a(n) = smallest m such that A049108(m)=n.

Alternatively, a(n) = smallest m such that A003434(m)=n-1.

a(n+2) ~ 2^n.

Extensions

More terms from David W. Wilson, May 15 1997

Additional comments from James S. Cronen (cronej(AT)rpi.edu)