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Irregular triangle of cycles of purely periodic unitary sigma aliquot sequences with their smallest member as starting number, read by rows.
3

%I #18 Jul 15 2021 21:26:36

%S 6,30,42,54,60,90,114,126,1140,1260,1482,1878,1890,2142,2178,2418,

%T 2958,3522,3534,4146,4158,3906,3774,4434,4446,3954,3966,3978,3582,

%U 18018,22302,24180,29580,35220,35340,41460,41580,39060,37740,44340,44460,39540,39660,39780,35820,32130,40446

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

%C 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 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 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 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 Irregular triangle of cycles:

%e 6

%e 30 42 54

%e 60

%e 90

%e 114 126

%e 1140 1260

%e 1482 1878 1890 2142 2178

%e 2418 2958 3522 3534 4146 4158 3906 3774 4434 4446 3954 3966 3978 3582

%e 18018 22302

%e ...

%t a063919[1] = 1; a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]] - n/;n>1 (* _Jean-François Alcover_ *)

%t aliquotSequence[n_] := NestWhileList[a063919, n, UnsameQ, All]

%t 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 Flatten[a336216[35000]] (* data - first 11 rows of triangle *)

%Y Cf. A034448, A063919, A327157, A336218, A336219.

%K nonn,tabf

%O 1,1

%A _Hartmut F. W. Hoft_, Jul 12 2020