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A097033
Number of transient terms before either 0 or a finite cycle is reached when unitary-proper-divisor-sum-function f(x) = A034460(x) is iterated and the initial value is n.
6
1, 2, 2, 2, 2, 0, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 2, 4, 2, 4, 3, 5, 2, 4, 2, 3, 2, 4, 2, 0, 2, 2, 4, 5, 3, 5, 2, 6, 3, 5, 2, 0, 2, 3, 4, 4, 2, 5, 2, 5, 4, 5, 2, 0, 3, 3, 3, 3, 2, 0, 2, 6, 3, 2, 3, 2, 2, 6, 3, 7, 2, 5, 2, 6, 3, 5, 3, 1, 2, 6, 2, 4, 2, 6, 3, 5, 5, 5, 2, 0, 4, 5, 4, 6, 3, 6, 2, 6, 4, 1, 2, 1, 2, 6, 6
OFFSET
1,2
LINKS
FORMULA
a(n) = A318883(n) + (1-A318880(n)). - Antti Karttunen, Sep 23 2018
EXAMPLE
From Antti Karttunen, Sep 24 2018: (Start)
For n = 1, A034460(1) = 0 that is, we end to a terminal zero after a transient part of length 1, thus a(1) = 1.
For n = 2, A034460(2) = 1, and A034460(1) = 0, so we end to a terminal zero after a transient part of length 2, thus a(2) = 2.
For n = 30, A034460(30) = 42, A034460(42) = 54, A034460(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. (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) = 6.
If a(n) = 0, then n is a term in an attractor set like A002827, A063991, A097024, A097030.
MATHEMATICA
a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/; n>0
transient[k_] := Module[{iter=NestWhileList[a034460, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]-1]
a097033[n_] := Map[transient, Range[n]]
a097033[105] (* Hartmut F. W. Hoft, Jan 24 2024 *)
PROG
(PARI)
A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A097033(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(mapget(visited, n)-1), mapput(visited, n, j)); n = A034460(n); if(!n, return(j))); }; \\ Antti Karttunen, Sep 23 2018
CROSSREFS
Cf. A318883 (sequence that implements the original definition of this sequence).
Sequence in context: A245476 A215884 A305029 * A268686 A113306 A181089
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 30 2004
EXTENSIONS
Definition corrected (to agree with the given terms) by Antti Karttunen, Sep 23 2018, based on observations by Hartmut F. W. Hoft
STATUS
approved