A096850 -id:A096850 - cp's OEIS frontend

A096852 a(n) is the length of terminal cycle of the trajectory of f(x)=phi(sigma(x)) if started at 2^n.

1, 1, 2, 1, 3, 2, 2, 1, 2, 2, 6, 2, 1, 6, 2, 1, 2, 3, 11, 11, 2, 2, 15, 15, 18, 18, 18, 18, 12, 12, 12, 1

Offset: 0

Labos Elemer, Jul 16 2004

nonn

			n=18: start = 262144 and the corresponding 11-cycle is 262144, 524286, [368640, 381024, 326592, 550368, 435456, 580608, 851840, 552960, 524160, 442368, 432000], 368640, ...

Cf. A095955, A095956, A096849, A096850.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := Block[{lst = NestWhileList[g, n, UnsameQ, All]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Table[ f[2^n], {n, 0, 20}]

f(x)=eulerphi(sigma(x))
a(n)=my(t=f(2^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 27 2013

a(n) = A095955(2^n). - Charles R Greathouse IV, Nov 27 2013

Edited, corrected and extended by Robert G. Wilson v, Jul 17 2004

A096849 If f(x) = phi(sigma(x)) is iterated starting from these numbers, then the start-value never returns. These are the transient terms of this iteration. Never occur in terminal cycles.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

Labos Elemer, Jul 16 2004

nonn

			All odd and certain even integers belong here.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A095952, A095953, A095954, A095955, A095956, A096887, A096888, A096889, A096890, A096850.

Flatten@ Table[Function[s, If[Length@ # > 0, First@ #, #] &@ Keys@ KeySelect[s, Length@ Lookup[s, #] == 1 &]]@ PositionIndex@ NestList[EulerPhi@ DivisorSigma[1, #] &, n, 10^2], {n, 71}] (* Michael De Vlieger, Jul 24 2017 *)

A096860 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in the transient of this trajectory, that is: a(n) = A096859(n) - A095955(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 0, 2, 1, 1, 4, 2, 3, 1, 3, 3, 1, 1, 0, 1, 3, 1, 2, 1, 1, 3, 2, 3, 1, 1, 0, 2, 2, 1, 1, 3, 1, 3, 1, 2, 2, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 3, 1, 3, 0, 4, 2, 1, 5, 3, 1, 1, 1, 3

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=255: list={255,144,360,288,[432,480],432,...}, t=transient=4, c=cycle=2, a(255)=t=4;
n=244: list={244,180,144,360,288,[432,480],432,...}, a(244)=4.
a(n)=0 means that n is a recurrent term from A096850.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A000203, A062401, A062402, A095955, A096859, A096861, A096865.

Cf. also A096863.

With[{nn = 120}, Array[Length@ Union@ # - Length@ Select[Tally@ #, Last@ # > 1 &] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, #, nn] &, 105]] (* Michael De Vlieger, Nov 18 2017 *)

(define (A096860 n) (let loop ((visited (list n))) (let ((next (A062401 (car visited)))) (cond ((member next visited) => (lambda (transientplusone) (- (length transientplusone) 1))) (else (loop (cons next visited))))))) ;; Antti Karttunen, Nov 18 2017

A096851 Even transient values of f(x)=phi(sigma(x)) iterations.

Original entry on oeis.org

10, 14, 18, 20, 22, 26, 28, 32, 34, 36, 38, 40, 42, 44, 46, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 1

Labos Elemer, Jul 16 2004

nonn

Michel Marcus, Table of n, a(n) for n = 1..4984

Even terms of A096849.

Cf. A095952-A095956, A096887-A096890, A096849-A096850.

A096998 Consider iteration of the function f(x) = sigma(phi(x)) = A062402(x). Sequence lists the numbers k such that the trajectory of k returns to k.

Original entry on oeis.org

1, 3, 7, 12, 15, 28, 31, 60, 72, 124, 168, 195, 252, 255, 744, 1240, 1512, 1651, 2418, 2520, 3066, 3844, 4092, 4800, 5080, 5376, 6045, 6138, 6552, 9906, 9920, 10200, 12264, 20440, 30855, 36792, 46228, 58968, 60984, 65535, 67963, 81880, 122640

Offset: 1

Labos Elemer, Jul 19 2004

nonn

			96 => 63 => 91 => 195 => 252 => 195 => ..., therefore 195 and 252 are in the sequence.

Cf. A062402, A096850.

a = {}; f[n_] := DivisorSigma[1, EulerPhi[ n]]; Do[ AppendTo[ a, NestWhileList[f, n, UnsameQ, All][[ -1]]]; a = Union[a], {n, 10^6}]; Take[ a, 43] (* Robert G. Wilson v, Jul 21 2004 *)

Edited, corrected and extended by Robert G. Wilson v, Jul 21 2004

cp's OEIS Frontend

A096852 a(n) is the length of terminal cycle of the trajectory of f(x)=phi(sigma(x)) if started at 2^n.

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A096849 If f(x) = phi(sigma(x)) is iterated starting from these numbers, then the start-value never returns. These are the transient terms of this iteration. Never occur in terminal cycles.

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A096860 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in the transient of this trajectory, that is: a(n) = A096859(n) - A095955(n).

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A096851 Even transient values of f(x)=phi(sigma(x)) iterations.

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A096998 Consider iteration of the function f(x) = sigma(phi(x)) = A062402(x). Sequence lists the numbers k such that the trajectory of k returns to k.

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