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A261144
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Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).
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13
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1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42
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OFFSET
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1,2
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COMMENTS
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If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1, 2; squarefree and 2-smooth
1, 2, 3, 6; squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...
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MAPLE
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b:= proc(n) option remember; `if`(n=0, [1],
sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
end:
T:= n-> b(n)[]:
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MATHEMATICA
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primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten
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CROSSREFS
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Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
A072047 counts prime factors of squarefree numbers.
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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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