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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. 8
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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A129288 A083248 A360765 * A254836 A067672 A276710
KEYWORD
nonn
AUTHOR
Labos Elemer, Oct 31 2002
EXTENSIONS
Name corrected by Charles R Greathouse IV, Nov 29 2013
STATUS
approved

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Last modified June 19 16:49 EDT 2024. Contains 373503 sequences. (Running on oeis4.)