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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.

%I A095955 #32 Aug 03 2024 07:16:07
%S A095955 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,
%T A095955 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,
%U A095955 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
%N 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.
%C A095955 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.
%C A095955 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.
%C A095955 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
%H A095955 Charles R Greathouse IV, <a href="/A095955/b095955.txt">Table of n, a(n) for n = 1..10000</a>
%H A095955 Jud McCranie, <a href="/A095955/a095955.txt">Unknown cases < 725000000</a>
%H A095955 Jud McCranie, <a href="/A095955/a095955_2.txt">Details of various cycles</a>
%H A095955 <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>
%e A095955 Occurrences of cycle lengths if n <= 1000: {C1=110, C2=781, C3=36, C4=67, C5=0, C6=6, C7=0, ...}.
%t A095955 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 *)
%o A095955 (PARI) f(x)=eulerphi(sigma(x))
%o A095955 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
%Y A095955 Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095956.
%K A095955 nonn
%O A095955 1,4
%A A095955 _Labos Elemer_, Jul 13 2004