A095954 -id:A095954 - cp's OEIS frontend

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.

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, 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, 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

Offset: 1

Labos Elemer, Jul 13 2004

nonn

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.
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.
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

			Occurrences of cycle lengths if n <= 1000: {C1=110, C2=781, C3=36, C4=67, C5=0, C6=6, C7=0, ...}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jud McCranie, Unknown cases < 725000000
Jud McCranie, Details of various cycles
Index to sequences related to decomposition of primes in quadratic fields

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095956.

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 *)

f(x)=eulerphi(sigma(x))
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

A095956 a(n) is the smallest initial value if function f(x)=phi(sigma(x)) is iterated and the iteration ends in a cycle of length n.

Original entry on oeis.org

1, 4, 16, 324

Offset: 1

Labos Elemer, Jul 13 2004

a(6) = 784. a(11) = 76050. a(15) = 467856. a(18) = 401408.
a(n) is the least k such that A095955(k) = n. a(9) = 22011456. a(12) = 14531344. No more terms < 10^8. - David Wasserman, May 14 2007
a(5) <= 6634509269055173050761216000, see A095955. - Charles R Greathouse IV, Nov 22 2013
a(5) <= 278832689509653754675200. a(21) <= 723197776. All numbers < 254731536 reach a cycle of length 1, 2, 3, 4, 6, 9, 11, 12, 15 or 18. 254731536 does not reach a cycle after 20000 iterations. - Donovan Johnson, Dec 06 2013
a(5) <= 9215376914800. a(7) <= 50566357997109706752. a(8) <= 41009810800. a(10) <= 8052138062400. a(16) <= 5281622477557929264. a(19) <= 12153003733213997291274240000. a(22) <= 46355253084. a(23) <= 164171309187293459251200. a(29) <= 15235849366671151595520000. a(31) <= 12904041477133188557545144320. a(34) <= 2611389824976. a(53) <= 760237821375852770392965120. a(56) <= 11067064315112568913920. a(80) <= 271411311216414271400943943680. a(93) <= 167350414807467078768. a(167) <= 19716297663934955520. a(351) <= 307625014110191616000000. a(595) <= 828718488676293128378677798502400. - Hiroaki Yamanouchi, Sep 10 2014
If none of the terms < 723197776 in the file of unknown terms in A095955 lead to a cycle of length 21, then a(21) = 723197776. - Jud McCranie, Jun 14 2024
a(8) = 41009810800, a(22) = 40941163200. - Jud McCranie, Jun 18 2024

			a(4) = 324. 324 -> 660 -> 576 -> 1512 -> 1280 -> 864 -> 576 (cycle length = 4). - _Donovan Johnson_, Dec 06 2013

Cf. A000010, A000203, A062401, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095955, A373739.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; t = Table[0, {50}]; Do[ d = f[n]; If[d < 51 && t[[d]] == 0, t[[d]] = n], {n, 1, 10^6}]; t (* Robert G. Wilson v, Jul 14 2004 *)

A095952 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=1.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 12, 14, 15, 19, 20, 22, 23, 26, 29, 41, 43, 128, 156, 168, 186, 189, 200, 201, 217, 231, 237, 240, 248, 254, 260, 266, 271, 285, 291, 297, 303, 304, 313, 314, 329, 332, 335, 337, 341, 346, 350, 366, 368, 383, 387, 395, 413, 427, 430, 436, 437

Offset: 1

Labos Elemer, Jul 13 2004

nonn

			n=100: trajectory={980, 648, 880, 720, 720}

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

Cf. A000010, A000203, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1540], f[ # ] == 1 &] (* Robert G. Wilson v, Jul 14 2004 *)

is(n)=my(t=f(n),h=f(t));while(t!=h,t=f(t);h=f(f(h)));t==f(t) \\ Charles R Greathouse IV, Nov 22 2013

A095953 Initial values for f(x) = phi(sigma(x)) such that iteration of f ends in a cycle of length 3.

Original entry on oeis.org

16, 18, 21, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 44, 45, 46, 47, 51, 53, 55, 58, 59, 61, 65, 71, 83, 86, 89, 109, 131, 137, 149, 900, 1116, 1152, 1156, 1200, 1236, 1260, 1300, 1320, 1380, 1386, 1410, 1428, 1458, 1488, 1500, 1518, 1524, 1533, 1536

Offset: 1

Labos Elemer, Jul 13 2004

nonn

			n=900: trajectory={900, 2160, 1920, [1536, 1200, 1860], 1536, ...}.

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

Cf. A000010, A000203, A095952, A096887, A096526, A095954, A096888, A096889, A096890, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1540], f[ # ] == 3 &] (* Robert G. Wilson v, Jul 14 2004 *)
fcl3Q[n_]:=Length[FindTransientRepeat[NestList[EulerPhi[DivisorSigma[1,#]]&,n,100],3][[2]]]==3; Select[Range[1600],fcl3Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 21 2016 *)

f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t==f(f(h)) \\ Charles R Greathouse IV, Nov 22 2013

More terms from Robert G. Wilson v, Jul 14 2004

A096887 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=2.

Original entry on oeis.org

4, 6, 7, 10, 11, 13, 17, 32, 36, 42, 48, 49, 50, 52, 54, 56, 57, 60, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114

Offset: 1

Labos Elemer and Robert G. Wilson v, Jul 14 2004

nonn

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

Cf. A000010, A000203, A095952, A095953, A096526, A095954, A096888, A096889, A096890, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[115], f[ # ] == 2 &]

f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t==f(h) \\ Charles R Greathouse IV, Nov 22 2013

A096890 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 18.

Original entry on oeis.org

401408, 414050, 436032, 455625, 462400, 466608, 476100, 486300, 486900, 512337, 522242, 526974, 543600, 544644, 544944, 546192, 546861, 554304, 559504, 571536, 572313, 575028, 577200, 579856, 583200, 585528, 599694, 604300, 609429, 611618

Offset: 1

Labos Elemer and Robert G. Wilson v, Jul 14 2004

nonn

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

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[624900], f[ # ] == 18 &]

f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); for(i=1,17,h=f(h); if(t==h,return(0))); t==f(h) \\ Charles R Greathouse IV, Nov 25 2013

A096888 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=11.

Original entry on oeis.org

76050, 88452, 88896, 95922, 104364, 104988, 106032, 108252, 110450, 110928, 112896, 113052, 113412, 113868, 115572, 119892, 119916, 121680, 122220, 122832, 122916, 123060, 123312, 129600, 129840, 129984, 130260, 130560, 131280, 133860

Offset: 1

Labos Elemer and Robert G. Wilson v, Jul 14 2004

nonn

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

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096889, A096890, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[134000], f[ # ] == 11 &]

f(x,k=1)=for(i=1,k,x=eulerphi(sigma(x))); x
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t==f(h,10) \\ Charles R Greathouse IV, Nov 25 2013

A096889 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=15.

Original entry on oeis.org

467856, 480200, 490000, 499968, 514416, 523344, 524352, 531441, 548400, 549444, 550512, 553728, 556752, 560532, 562500, 562800, 570000, 570276, 570576, 573744, 576240, 579024, 579900, 581700, 584112, 584836, 586756, 590268, 595008, 599076

Offset: 1

Labos Elemer and Robert G. Wilson v, Jul 14 2004

nonn

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

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096890, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[599800], f[ # ] == 15 &]

f(x,k=1)=for(i=1,k,x=eulerphi(sigma(x))); x
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t!=(h=f(h,2)) && t!=(h=f(h,2)) && t==f(h,10) \\ Charles R Greathouse IV, Nov 25 2013

A096526 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 4.

Original entry on oeis.org

324, 400, 484, 490, 530, 544, 576, 630, 660, 672, 684, 690, 714, 722, 750, 756, 768, 770, 772, 777, 780, 792, 804, 810, 819, 828, 832, 833, 840, 841, 846, 852, 858, 864, 868, 870, 872, 876, 888, 892, 901, 906, 910, 918, 920, 924, 930, 936, 940, 948, 952

Offset: 1

Labos Elemer, Jul 13 2004

nonn

			n=324: trajectory=324, 660, [576, 1512, 1280, 864], 576, ...}

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

Cf. A000010, A000203, A095952, A096887, A095953, A095954, A096888, A096889, A096890, A095955, A095956.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1000], f[ # ] == 4 &] (* Robert G. Wilson v, Jul 14 2004 *)

f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t!=(h=f(h)) && t==f(f(h)) \\ Charles R Greathouse IV, Nov 22 2013

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 *)

cp's OEIS Frontend

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.

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A095956 a(n) is the smallest initial value if function f(x)=phi(sigma(x)) is iterated and the iteration ends in a cycle of length n.

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A095952 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=1.

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A095953 Initial values for f(x) = phi(sigma(x)) such that iteration of f ends in a cycle of length 3.

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A096887 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=2.

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A096890 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 18.

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A096888 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=11.

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A096889 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=15.

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A096526 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 4.

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