A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.

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

Offset: 1

Author

Maghraoui Abdelkader

Keywords

• nonn
• easy
• nice
• tabf

Comments

n-th row has length A001222(n) (n>1).

Examples

			Triangle begins
  1;
  2;
  3;
  2, 2;
  5;
  2, 3;
  7;
  2, 2, 2;
  3, 3;
  2, 5;
  11;
  2, 2, 3;
  ...
		

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.

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.

Programs

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

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 *)

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

Python

def factors(n: int) -> list[int]:
    p = n
    L:list[int] = []
    for f in range(2, p + 1):
        if f * f > p: break
        while True:
            q, r = divmod(p, f)
            if r != 0: break
            L.append(f)
            p = q
            if p == 1: return L
    L.append(p)
    return L  # Peter Luschny, Jul 18 2024

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

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)

Extensions

More terms from James Sellers