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%I #16 Jan 29 2024 13:49:02
%S 1,2,2,2,2,1,2,2,2,3,2,3,2,4,3,2,2,4,2,4,3,5,2,4,2,3,2,4,2,3,2,2,4,5,
%T 3,5,2,6,3,5,2,3,2,3,4,4,2,5,2,5,4,5,2,3,3,3,3,3,2,1,2,6,3,2,3,3,2,6,
%U 3,7,2,5,2,6,3,5,3,2,2,6,2,4,2,6,3,5,5,5,2,1,4,5,4,6,3,6,2,6,4,4,2,3,2,6,6
%N Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n. Number of distinct terms in iteration list.
%C This sequence implements the original definition given for A097032.
%H Antti Karttunen, <a href="/A318882/b318882.txt">Table of n, a(n) for n = 1..87360</a>
%F a(n) = A097031(n) + A318883(n).
%F a(n) = A097032(n) + A318880(n) - 1.
%e 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) = 0+1 = 1.
%e 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+1 = 2.
%e For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(54) = 30, thus a(30) = a(42) = a(54) = 0+3 = 3, as 30, 42 and 54 are all contained in their own terminal cycle of length 3, without a preceding transient part.
%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+14 = 20.
%t a063919[1] = 1; (* function a[] in A063919 by _Jean-François Alcover_ *)
%t a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
%t a318882[n_] := Map[Length[NestWhileList[a063919, #, UnsameQ, All]]-1&, Range[n]]
%t a318882[105] (* _Hartmut F. W. Hoft_, Jan 25 2024 *)
%o (PARI)
%o A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
%o A063919(n) = if(1==n,n,A034460(n));
%o A318882(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-1), mapput(visited, n, j)); n = A063919(n)); };
%o \\ Or by using lists:
%o pil(item,lista) = { for(i=1,#lista,if(lista[i]==item,return(i))); (0); };
%o A318882(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(j-1)); listput(visited, n); n = A063919(n)); };
%Y Cf. A063919, A097031, A097032, A318883.
%Y Cf. A002827 (the positions of ones after the initial 1).
%K nonn
%O 1,2
%A _Antti Karttunen_, Sep 22 2018, after _Labos Elemer_'s A097032
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