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
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
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 13 2004
STATUS
approved