A095955 Function f(x) = phi(sigma(x)) is iterated with initial value n; a(n) is the length of the cycle into which the trajectory merges.

1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 3, 1, 1, 3, 1, 1, 3, 3, 1, 3, 3, 1, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 1, 2, 1, 3, 3, 3, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,4

Comments

Diagnosis of true cycle of length m: a(j-m) = a(j), but a(j-d) = a(j) cases are excluded for d dividing m.

Length 5 is rare. Example: a(6634509269055173050761216000)=5 and the 5-cycle is {6634509269055173050761216000, 7521613519844726223667200000, 7946886558074859593662464000, 7794495412499746337587200000, 7970172471593905204651622400, 6634509269055173050761216000}. The initial values 2^79 = 604462909807314587353088 and 2^83 = 9671406556917033397649408 after more than 250 transient terms reach this cycle.

a(i) is in {1,2,3,4,6,9,11,12,15,18} for 1 <= i < 254731536. The number 254731536 is the smallest of many integers that are not known to reach a cycle (see the file for a list). - Jud McCranie, Jun 05 2024

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Jud McCranie, Unknown cases < 725000000

Jud McCranie, Details of various cycles

Index to sequences related to decomposition of primes in quadratic fields

Example

Occurrences of cycle lengths if n <= 1000: {C1=110, C2=781, C3=36, C4=67, C5=0, C6=6, C7=0, ...}.

Mathematica

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jul 14 2004 *)

Prog

(PARI) f(x)=eulerphi(sigma(x))
a(n)=my(t=f(n), h=f(t), s); while(t!=h, t=f(t); h=f(f(h))); t=f(t); h=f(t); s=1; while(t!=h, s++; t=f(t); h=f(f(h))); s \\ Charles R Greathouse IV, Nov 22 2013

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095956.

Sequence in context: A207676 A161175 A356515 * A272772 A293431 A078573

Adjacent sequences: A095952 A095953 A095954 * A095956 A095957 A095958

Keyword

nonn

Author

Labos Elemer, Jul 13 2004

Status

approved