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

Author

Labos Elemer, Jul 13 2004

Keywords

• nonn

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

Examples

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

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

Crossrefs

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

Programs

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 *)

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