A077090 When iteration of f(x) = phi(sigma(x) - phi(x)) is started at initial values listed here it ends up in a cycle of length greater than 1.

36, 40, 48, 50, 52, 60, 64, 66, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 100, 102, 105, 106, 108, 110, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126, 128, 130, 132, 135, 136, 138, 140, 141, 142, 144, 146, 147, 148, 150, 152, 154, 156, 159, 160, 162, 164

Offset

1,1

Comments

Below 2000 there are only these two cycles of length 3: 36, 78, 48; and 144, 280, 192.

For most composite numbers below 80, the iterated trajectory of f settles on a single-number cycle of 4; those numbers are not in this sequence. - Alonso del Arte, Nov 29 2013

Links

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

Example

36 is in the sequence because f(36) = 78, f(78) = 48 and f(48) = 36, which is a cycle of length 3.
38 is not in the sequence because iterating f from 38 gives the trajectory 38, 12, 8, 10, 6, 4, 4, 4, ... where the cycle has a length of 1.

Mathematica

f[x_] := EulerPhi[DivisorSigma[1, x] - EulerPhi[x]]; Do[s = NestList[f, n, 100]; s1 = Part[s, 98]; s2 = Part[s, 99]; s3 = Part[s, 100]; If[ !Equal[s1, s2], k = k + 1; Print[{k, n, s1, s2, s3}]], {n, 2, 1000}]
f[n_] := EulerPhi[DivisorSigma[1, n]-EulerPhi[n]]; cycleQ[n_] := With[{nn = NestWhileList[f, n, Unequal, All]}, nn[[-1]] != nn[[-2]]]; Select[Range[200], cycleQ] (* Jean-François Alcover, Nov 29 2013 *)

Prog

(PARI) f(x)=if(x>35, eulerphi(sigma(x)-eulerphi(x)), 1)
is(n)=my(t=f(n), h=f(t)); while(t!=h, h=f(f(h)); t=f(t)); f(t)!=t \\ Charles R Greathouse IV, Nov 29 2013

Crossrefs

Cf. A077088, A077091-A077100.

Sequence in context: A129288 A083248 A360765 * A254836 A067672 A276710

Adjacent sequences: A077087 A077088 A077089 * A077091 A077092 A077093

Keyword

nonn

Author

Labos Elemer, Oct 31 2002

Extensions

Name corrected by Charles R Greathouse IV, Nov 29 2013

Status

approved