A096857 - OEIS

%I #10 Jan 09 2024 12:13:04

%S 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,

%T 12,12,1

%N a[n] is the length of terminal cycle of the trajectory of g[x]=sigma(phi(x)) if started at 2^n. Formally identical to A096852, but arguments are shifted by 1 and the iterated functions are different!.

%C Offset=1 in contrast to A096852, where offset=0. Also the iterated functions deviate: A062401 iterated in A096852 and A062402 is repeated here; A096852(n)=A096857(n+1) appears to be true. While cycle-lengths seem identical, the composition of cycles are mostly different!

%e n=5:iv=32 list={32,[31,72,60]} length=a(5)=3, while the parallel case of A096852(n)=b(n) is b[4] with [16,24,30] cycle.

%e Also A096857[11] starts with 2048 ends in 6-cycle: {2048,2047,4123,10890,8928,[9906,9920,12264,10200,6138,6045],9906,..

%e while A096852[11-1]=6 and the relevant 6-cycle is {1024,1936,3240,2640,[2880,3024,3840,3456,2560,1800],2880,... These are different cycles with identical lengths.

%e The initial value 146 leads to list with enormous terms.

%t f[n_] := DivisorSigma[1, EulerPhi[n]]; g[n_] := Block[{l = NestWhileList[f, 2^n, UnsameQ, All]}, -Subtract @@ Flatten[Position[l, l[[ -1]]]]]; Table[ g[n], {n, 25}] (* _Robert G. Wilson v_, Jul 21 2004 *)

%Y Cf. A062401, A062402, A096852, A096858.

%K nonn

%O 1,3

%A _Labos Elemer_, Jul 19 2004