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Search: id:A112682
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| A112682 |
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Triangular table read by rows: T[n,k] counts the occurrences of integer k in the sequence generated by replacing integer i by the sorted sequence of divisors of (i+1), starting on 1 and iterating n times. |
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+0
7
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| 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 5, 3, 1, 1, 19, 13, 6, 3, 1, 1, 43, 26, 14, 6, 3, 1, 1, 94, 61, 29, 15, 6, 3, 1, 1, 210, 130, 68, 30, 15, 6, 3, 1, 1, 464, 297, 146, 71, 31, 15, 6, 3, 1, 1, 1035, 648, 331, 152, 72, 31, 15, 6, 3, 1, 1, 2295, 1457, 727, 347, 155, 73, 31, 15, 6, 3, 1, 1
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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T[n+1,1] = sum of previous row (each integer produces the trivial divisor 1). The matrix inverse M (after padding to the right with zero's) consists of columns that, from the main diagonal down, are nested Shift-Moebius Transforms of {1,0,0,..0} (see %t line).
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EXAMPLE
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The linear substitution sequence is:
1
1,2
1,2,1,3
1,2,1,3,1,2,1,2,4
1,2,1,3,1,2,1,2,4,1,2,1,3,1,2,1,3,1,5
(* limiting sequence is eigenfunction of the operator *)
Counting each of the integers results in:
{1},
{1,1},
{2,1,1},
{4,3,1,1},
{9,5,3,1,1}
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MATHEMATICA
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(Length/@ Split[Sort[ # ]])&/@ NestList[Flatten[ #/. k_:>Divisors[1+k]]&, {1}, 12]; or, more efficiently: Nest[Apply[Plus, Map[Last, Split[Sort[Apply[Sequence, Thread[w[Divisors[1 +Range[Length[ # ]]]& @ #, List/@# ]]/. w->(Outer[Sequence, ## ]&), {1}]], First[ #1]===First[ #2]&], {2}], {1}]&, {1}, 63]; or, using a Shift-Moebius Transform: upper=MapIndexed[Drop[ #1, -1+First@#2]&, IdentityMatrix[17], {1}]; tran=Rest/@ MapIndexed[Nest[ Prepend[moebius[ # ], 0]&, #1, First@#2]&, upper]; MapIndexed[Take[ #1, First@#2]&, Transpose[Inverse[tran]], {1}]
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CROSSREFS
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Cf. A008683.
Sequence in context: A103323 A092056 A103574 this_sequence A033185 A105632 A091491
Adjacent sequences: A112679 A112680 A112681 this_sequence A112683 A112684 A112685
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KEYWORD
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tabl,nonn
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AUTHOR
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Wouter Meeussen (wouter.meeussen(AT)pandora.be) and Paul D. Hanna (pauldhanna(AT)juno.com), Dec 31 2005s; revised Jan 23 2006
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