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.

1, 4, 16, 324

Offset: 1

Author

Labos Elemer, Jul 13 2004

Keywords

• nonn
• hard
• more

Comments

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

Examples

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

Crossrefs

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

Programs

Mathematica

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 *)