A097032 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A034460(x) is iterated and the initial value is n. Number of distinct terms in iteration list, including also the terminal 0 in the count if the iteration doesn't end in a cycle.

2, 3, 3, 3, 3, 1, 3, 3, 3, 4, 3, 4, 3, 5, 4, 3, 3, 5, 3, 5, 4, 6, 3, 5, 3, 4, 3, 5, 3, 3, 3, 3, 5, 6, 4, 6, 3, 7, 4, 6, 3, 3, 3, 4, 5, 5, 3, 6, 3, 6, 5, 6, 3, 3, 4, 4, 4, 4, 3, 1, 3, 7, 4, 3, 4, 3, 3, 7, 4, 8, 3, 6, 3, 7, 4, 6, 4, 2, 3, 7, 3, 5, 3, 7, 4, 6, 6, 6, 3, 1, 5, 6, 5, 7, 4, 7, 3, 7, 5, 4, 3, 3, 3, 7, 7

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = A318882(n) + (1-A318880(n)). - Antti Karttunen, Sep 23 2018

Example

From Antti Karttunen, Sep 24 2018: (Start)
For n = 1, A034460(1) = 0, thus a(1) = 1+1 = 2.
For n = 2, A034460(2) = 1, and A034460(1) = 0, so we end to the zero after a transient part of length 2, thus a(2) = 2+1 = 3.
For n = 30, A034460(30) = 42, A034460(42) = 54, A034460(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. (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+14 = 20.

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
a097032[n_] := Map[Length[NestWhileList[a034460, #, UnsameQ, All]]-1&, Range[n]]
a097032[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
A097032(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-1), mapput(visited, n, j)); n = A034460(n); if(!n, return(j+1))); }; \\ Antti Karttunen, Sep 23 2018

Crossrefs

Cf. A003062, A034460, A063919, A063991, A097024, A097030, A097031, A097033-A097037, A318880, A318882.

Cf. A002827 (the positions of ones).

Cf. A318882 (sequence that implements the original definition of this sequence).

Sequence in context: A322225 A110049 A246577 * A127661 A358617 A008968

Adjacent sequences: A097029 A097030 A097031 * A097033 A097034 A097035

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