login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Maghraoui Abdelkader

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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