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 A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition. 142
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS n-th row has length A001222(n) (n>1). LINKS N. J. A. Sloane, First 2048 rows of triangle, flattened S. von Worley (?), Animated Factorization Diagrams, Oct. 2012. Brent Yorgey, Factorization diagrams, The Math Less Traveled, Oct 05 2012. FORMULA Product_{k=1..A001222(n)} T(n,k) = n. From Reinhard Zumkeller, Aug 27 2011: (Start) A001414(n) = Sum_{k=1..A001222(n)} T(n,k), n>1; A006530(n) = T(n,A001222(n)) = Max_{k=1..A001222(n)} T(n,k); A020639(n) = T(n,1) = Min_{k=1..A001222(n)} T(n,k). (End) EXAMPLE Triangle begins 1; 2; 3; 2, 2; 5; 2, 3; 7; 2, 2, 2; 3, 3; 2, 5; 11; 2, 2, 3; ... MAPLE 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 MATHEMATICA row[n_] := Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Flatten[ Table[ row[n], {n, 1, 45}]] (* Jean-François Alcover, Dec 01 2011 *) PROG (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 -- Reinhard Zumkeller, Aug 27 2011 (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)) print(v) # Giuseppe Coppoletta, Dec 29 2017 CROSSREFS Cf. A000027, A001222, A027748. a(A022559(A000040(n))+1) = A000040(n). Column 1 is A020639, columns 2 and 3 correspond to A014673 and A115561. A281890 measures frequency of each prime in each column, with A281889 giving median values. Cf. A175943 (partial products), A265110 (partial row products), A265111. Sequence in context: A336526 A225243 A207338 * A307746 A348477 A240230 Adjacent sequences: A027743 A027744 A027745 * A027747 A027748 A027749 KEYWORD nonn,easy,nice,tabf AUTHOR EXTENSIONS More terms from James A. Sellers STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)