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A336216 Irregular triangle of cycles of purely periodic unitary sigma aliquot sequences with their smallest member as starting number, read by rows.

%I A336216 #18 Jul 15 2021 21:26:36
%S A336216 6,30,42,54,60,90,114,126,1140,1260,1482,1878,1890,2142,2178,2418,
%T A336216 2958,3522,3534,4146,4158,3906,3774,4434,4446,3954,3966,3978,3582,
%U A336216 18018,22302,24180,29580,35220,35340,41460,41580,39060,37740,44340,44460,39540,39660,39780,35820,32130,40446
%N A336216 Irregular triangle of cycles of purely periodic unitary sigma aliquot sequences with their smallest member as starting number, read by rows.
%C A336216 For the definition of unitary divisors see A034448. This sequence is a permutation of A327157; the starting numbers of successive cycles are in increasing order; the numbers in a cycle are kept in the order of the iteration with the smallest number in the cycle as the starting number. In order to be consistent with A327157 the terminal 1-cycle consisting of 1 is not included in the sequence.
%C A336216 Sequence A336218 gives the cycle lengths, therefore the start of the k-th cycle in this sequence is at index 1 + Sum_{i=1..k-1} A336218(i). Sequence A336219 is the first column of the triangle.
%C A336216 From the formula of _Vladeta Jovovic_ in A034448, it follows that all unitary aliquot sequences, and hence cycles, contain only odd numbers or only even numbers (except for the possible terminal 1). The table of _Antti Karttunen_ in the link of A327157 includes just 2 odd cycles, the 2-cycles: 8619765, 9627915 and 17257695, 17578785.
%e A336216 The first cycle of size 14 starting at position 16 is: 2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582. Its 7th element is the first number in this sequence smaller than its predecessor.
%e A336216 Irregular triangle of cycles:
%e A336216 6
%e A336216 30    42   54
%e A336216 60
%e A336216 90
%e A336216 114   126
%e A336216 1140  1260
%e A336216 1482  1878 1890 2142 2178
%e A336216 2418  2958 3522 3534 4146 4158 3906 3774 4434 4446 3954 3966 3978 3582
%e A336216 18018 22302
%e A336216 ...
%t A336216 a063919[1] = 1; a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]] - n/;n>1 (* _Jean-François Alcover_ *)
%t A336216 aliquotSequence[n_] := NestWhileList[a063919, n, UnsameQ, All]
%t A336216 a336216[n_] := Module[{list={}, listS={}, i, seq, seqS}, For[i=2, i<=n, i++, seq=aliquotSequence[i]; If[First[seq]==Last[seq], seqS=Sort[Most[seq]]; If[!MemberQ[listS, seqS], AppendTo[listS, seqS]; AppendTo[list, Most[seq]]]]]; list] (* list of cycles *)
%t A336216 Flatten[a336216[35000]] (* data - first 11 rows of triangle *)
%Y A336216 Cf. A034448, A063919, A327157, A336218, A336219.
%K A336216 nonn,tabf
%O A336216 1,1
%A A336216 _Hartmut F. W. Hoft_, Jul 12 2020