This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
n=255: list={255,144,360,288,[432,480],432,...}, t=transient=4, c=cycle=2, a(255)=t+c=6;
n=244: list={244,180,144,360,288,[432,480],432,...}, t=5, c=2, a(244)=7.
fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 20]]], {w, 1, 256}] (* len=20 at n<=256 is suitable *)
(define (A096859 n) (let loop ((visited (list n)) (i 1)) (let ((next (A062401 (car visited)))) (cond ((member next visited) i) (else (loop (cons next visited) (+ 1 i))))))) ;; Antti Karttunen, Nov 18 2017
n=255: list={255,144,360,288,[432,480],432,...}, a(255)=144 as a transient term;
n=254: list={254,[128],128,...}, a(254)=128, as a fixed point.
fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, hos_] :=NestList[fs, x, hos] Table[Min[itef[w, 20]], {w, 1, 256}]
(define (A096865 n) (let loop ((visited (list n)) (m n)) (let ((next (A062401 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (min m next))))))) ;; Antti Karttunen, Nov 18 2017
Examples of cycles: {[1], [2], [4, 6], [8], [12], [16, 30, 24], [48, 60], [72, 96], [128]}.
95 => 32 => 36 => 72 => 96 => 72 => ..., therefore 72 and 96 are in the sequence.
a = {}; f[n_] := EulerPhi[ DivisorSigma[ 1, n]]; Do[ AppendTo[a, NestWhileList[f, n, UnsameQ, All][[ -1]]]; a = Union[a], {n, 10^6}]; Take[ a, 46] (* Robert G. Wilson v, Jul 21 2004 *)
f(n)=eulerphi(sigma(n)) is(n)=my(t=f(n),h=f(t));while(t!=h,t=f(t);h=f(f(h));if(t==n, return(1)));t==n \\ Charles R Greathouse IV, Nov 27 2013
n=255: list={255,144,360,288,[432,480],432,...}, a(255)=480, a recurrent term;
n=247: list={247,96,72,96,...}, a(247)=247, a transient term, here the initial value.
gf[x_] :=DivisorSigma[1, EulerPhi[x]] itef[x_, len_] :=NestList[fs, x, len] Table[Max[itef[w, 20]], {w, 1, 256}]
(define (A096861 n) (let loop ((visited (list n)) (m n)) (let ((next (A062401 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (max m next))))))) ;; Antti Karttunen, Nov 18 2017
With[{s = Array[EulerPhi@ DivisorSigma[1, #] &, 10^5]}, Union@ FoldList[Max, s] ] (* Michael De Vlieger, Dec 06 2018 *)
{ n=r=0; for (m=1, 10^9, x=eulerphi(sigma(m)); if (x > r, r=x; write("b065392.txt", n++, " ", x); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009
n=0: trajectory = {1,1,..} so a(0)=0;
n=14: transient-length=14, cycle-length=2, a(14)=14, A096852(14)=2; trajectory ={16384, 27000, 23040, 21600, 17280, 15360, 15488, 13824, 9600, 7680, 7200, 12960, 11880, 11520, [10368,14080], 10368, ...}.
Values of a(n) for n > 31, with -1 signifying transient lengths yet unknown after 10^4 iterations of f(x): -1, 7, 51, 70, 23, 39, 11, -1, 37, 107, 30, -1, 145, 25, 21, 36, -1, -1, -1, -1, 31, -1, 452, -1, 449, 447, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 40, -1, -1, -1, -1, -1, -1, -1, 279, -1. - _Michael De Vlieger_, May 15 2017
With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ #, k_ /; Length@ k == 1] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, 2^n, nn] /. k_ /; k == nn + 1 -> -1, {n, 31}] ] (* Michael De Vlieger, May 15 2017, Version 10 *)
Initial segment of A062401: {1, 2, 2, 6, 2, 4, 4, 8, 12, 6, 4, 12, 6, 8, 8, 30, 6, ...}. The peak values (those exceeding all previous ones) are 1, 2, 6, 8, 12, 30, reached at positions 1, 2, 4, 8, 9, 16, respectively.
a = 0; s = 0; Do[s = EulerPhi[DivisorSigma[1, n]]; If[s > a, a = s; Print[n]], {n, 1, 10^6}]
(* Second program: *)
With[{s = Array[EulerPhi@ DivisorSigma[1, #] &, 2*10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Dec 06 2018 *)
DeleteDuplicates[Table[{n,EulerPhi[DivisorSigma[1,n]]},{n,150000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[;;,1]] (* Harvey P. Dale, May 12 2023 *)
{ n=r=0; for (m=1, 10^9, x=eulerphi(sigma(m)); if (x > r, r=x; write("b065391.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009
Table[ EulerPhi[ DivisorSigma[1, 2^n - 1]], {n, 33}]
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