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A318883
Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.
6
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
1,10
COMMENTS
This sequence implements the original definition given for A097033.
LINKS
FORMULA
a(n) = A318882(n) - A097031(n).
a(n) = A097033(n) + A318880(n) - 1.
EXAMPLE
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.
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 *)
PROG
(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)); };
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Sep 22 2018, after Labos Elemer's A097033
STATUS
approved