A318883 Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 3, 2, 4, 1, 3, 1, 2, 1, 3, 1, 0, 1, 1, 3, 4, 2, 4, 1, 5, 2, 4, 1, 0, 1, 2, 3, 3, 1, 4, 1, 4, 3, 4, 1, 0, 2, 2, 2, 2, 1, 0, 1, 5, 2, 1, 2, 2, 1, 5, 2, 6, 1, 4, 1, 5, 2, 4, 2, 1, 1, 5, 1, 3, 1, 5, 2, 4, 4, 4, 1, 0, 3, 4, 3, 5, 2, 5, 1, 5, 3, 1, 1, 1, 1, 5, 5

Offset: 1

Author

Antti Karttunen, Sep 22 2018, after Labos Elemer's A097033

Keywords

• nonn

Comments

This sequence implements the original definition given for A097033.

Examples

			For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle (of length 1 in this case), thus there are no transient part, and a(1) = 0.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle 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) = 0, as 30, 42 and 54 are all contained in their own terminal cycle, without a preceding transient part.
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) = 6.
If a(n) = 0, then n is a term in an attractor set like A002827, A063991, A097024, A097030.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..87360

Crossrefs

Cf. A003062, A063919, A097031, A097033, A318880, A318882.

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
transient[k_] := Module[{iter=NestWhileList[a063919, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]]-1
a318883[n_] := Map[transient, Range[n]]
a318883[105] (* Hartmut F. W. Hoft, Jan 25 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));
A318883(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(mapget(visited, n)-1), 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); };
A318883(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(k-1)); listput(visited, n); n = A063919(n)); };

Formula

a(n) = A318882(n) - A097031(n).

a(n) = A097033(n) + A318880(n) - 1.