A097031 Length of terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Author

Labos Elemer, Aug 30 2004

Keywords

• nonn

Examples

			From _Antti Karttunen_, Sep 22 2018: (Start)
For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle of length 1 without a preceding transient part, thus a(1) = 1.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle of length 1 (after a transient part of length 1) thus a(2) = 1.
For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(54) = 30, thus a(30) = a(42) = a(54) = 3, as 30, 42 and 54 are all contained in their own terminal cycle of length 3, without a preceding transient part. (End)
For n = 1506, the iteration-list is {1506, 1518, 1938, 2382, 2394, 2406, [2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582, 2418, ..., ad infinitum]}. After a transient of length 6 the iteration ends in a cycle of length 14, thus a(1506) = 14.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A002827, A063919, A063991, A097024, A097030, A097032, A097033, A097034, A097035, A097036, A097037, A318882, A318883.

Programs

Mathematica

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
cycleLength[k_] := Module[{cycle=NestWhileList[a063919, k, UnsameQ, All]}, Apply[Subtract, Reverse[Flatten[Position[cycle, Last[cycle]], 1]]]]
a097031[n_] := Map[cycleLength, Range[n]]
a097031[105] (* Hartmut F. W. Hoft, Jan 24 2024 *)

PARI

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n));
A097031(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-mapget(visited, n)), mapput(visited, n, j)); n = A063919(n)); };
\\ Or by using lists:
pil(item,lista) = { for(i=1,#lista,if(lista[i]==item,return(i))); (0); };
A097031(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(j-k)); listput(visited, n); n = A063919(n)); }; \\ Antti Karttunen, Sep 22 2018

Formula

a(n) = A318882(n) - A318883(n). - Antti Karttunen, Sep 22 2018