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A027746
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Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.
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225
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1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 2, 2, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 2, 2, 2, 17, 2, 3, 3, 19, 2, 2, 5, 3, 7, 2, 11, 23, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 29, 2, 3, 5, 31, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 37, 2, 19, 3, 13, 2, 2, 2, 5, 41, 2, 3, 7, 43, 2, 2, 11, 3, 3, 5
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OFFSET
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1,2
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COMMENTS
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n-th row has length A001222(n) (n>1).
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LINKS
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FORMULA
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Product_{k=1..A001222(n)} T(n,k) = n.
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EXAMPLE
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Triangle begins
1;
2;
3;
2, 2;
5;
2, 3;
7;
2, 2, 2;
3, 3;
2, 5;
11;
2, 2, 3;
...
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MAPLE
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P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: 1; for n from 2 to 45 do P(n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 13 2005
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MATHEMATICA
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row[n_] := Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Flatten[ Table[ row[n], {n, 1, 45}]] (* Jean-François Alcover, Dec 01 2011 *)
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PROG
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(Haskell)
import Data.List (unfoldr)
a027746 n k = a027746_tabl !! (n-1) !! (k-1)
a027746_tabl = map a027746_row [1..]
a027746_row 1 = [1]
a027746_row n = unfoldr fact n where
fact 1 = Nothing
fact x = Just (p, x `div` p) where p = a020639 x
(PARI) A027746_row(n, o=[1])=if(n>1, concat(apply(t->vector(t[2], i, t[1]), Vec(factor(n)~))), o) \\ Use %(n, []) if you want the more natural [] for the first row. - M. F. Hasler, Jul 29 2015
(Sage) v=[1]
for k in [2..45]: v.extend(p for (p, m) in factor(k) for _ in range(m))
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CROSSREFS
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A281890 measures frequency of each prime in each column, with A281889 giving median values.
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KEYWORD
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nonn,easy,nice,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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