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A336216
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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.
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3
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6, 30, 42, 54, 60, 90, 114, 126, 1140, 1260, 1482, 1878, 1890, 2142, 2178, 2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582, 18018, 22302, 24180, 29580, 35220, 35340, 41460, 41580, 39060, 37740, 44340, 44460, 39540, 39660, 39780, 35820, 32130, 40446
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OFFSET
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1,1
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COMMENTS
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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.
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.
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.
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LINKS
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EXAMPLE
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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.
Irregular triangle of cycles:
6
30 42 54
60
90
114 126
1140 1260
1482 1878 1890 2142 2178
2418 2958 3522 3534 4146 4158 3906 3774 4434 4446 3954 3966 3978 3582
18018 22302
...
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MATHEMATICA
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a063919[1] = 1; a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]] - n/; n>1 (* Jean-François Alcover *)
aliquotSequence[n_] := NestWhileList[a063919, n, UnsameQ, All]
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 *)
Flatten[a336216[35000]] (* data - first 11 rows of triangle *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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