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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. 102
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).

A001414(n)=Sum(T(n,k):1<=k<=A001222(n)), n>1; A006530(n) = T(n,A001222(n)) = Max(T(n,k):1<=k<=A001222(n)); A020639(n) = T(n,1) = Min(T(n,k):1<=k<=A001222(n)). [Reinhard Zumkeller, Aug 27 2011]

LINKS

N. J. A. Sloane, First 2048 rows of triangle, flattened

S. von Worley (?), Animated Factorization Diagrams, Oct. 2012.

FORMULA

Product(T(n,k): 1 <= k <= A001221(n)) = n.

EXAMPLE

Triangle begins

  1;

  2;

  3;

  2, 2;

  5;

  2, 3;

  7;

  2, 2, 2;

  3, 3;

  2, 5;

  ...

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-Fran├žois 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=add(([p] * m for (p, m) in factor(k)), v)

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: A118665 A225243 A207338 * A240230 A238689 A166454

Adjacent sequences:  A027743 A027744 A027745 * A027747 A027748 A027749

KEYWORD

nonn,easy,nice,tabf

AUTHOR

Maghraoui Abdelkader

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified December 10 00:30 EST 2018. Contains 318032 sequences. (Running on oeis4.)