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%I #21 Jan 24 2024 10:24:14
%S 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,
%T 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,
%U 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
%N 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.
%H Antti Karttunen, <a href="/A097033/b097033.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A318883(n) + (1-A318880(n)). - _Antti Karttunen_, Sep 23 2018
%e From _Antti Karttunen_, Sep 24 2018: (Start)
%e 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.
%e 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.
%e 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)
%e 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.
%e If a(n) = 0, then n is a term in an attractor set like A002827, A063991, A097024, A097030.
%t a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
%t a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>0
%t transient[k_] := Module[{iter=NestWhileList[a034460, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]-1]
%t a097033[n_] := Map[transient, Range[n]]
%t a097033[105] (* _Hartmut F. W. Hoft_, Jan 24 2024 *)
%o (PARI)
%o A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
%o 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
%Y Cf. A002827, A003062, A034460, A063919, A063991, A097024, A097030, A097031, A097032-A097037, A318880.
%Y Cf. A318883 (sequence that implements the original definition of this sequence).
%K nonn
%O 1,2
%A _Labos Elemer_, Aug 30 2004
%E Definition corrected (to agree with the given terms) by _Antti Karttunen_, Sep 23 2018, based on observations by _Hartmut F. W. Hoft_
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