A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 4, 3, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 8, 3, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 8, 5, 41, 2, 3, 7, 43, 4, 11, 9, 5, 2, 23, 47, 16, 3, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 8, 7, 3

Offset: 1

Author

Leroy Quet, Jul 07 2008

Keywords

• nonn
• tabf

Comments

In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - Franklin T. Adams-Watters, May 05 2011

A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - Reinhard Zumkeller, Jan 29 2013

Examples

			60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts:
    n : largest prime powers dividing n
    1 :  1
    2 :  2
    3 :  3
    4 :  4
    5 :  5
    6 :  2, 3
    7 :  7
    8 :  8
    9 :  9
   10 :  2, 5
   11 : 11
   12 :  4, 3
   etc. (End)
		

Links

• Reinhard Zumkeller, Rows n=1..10000 of triangle, flattened
• Eric Weisstein's World of Mathematics, Prime Factorization

Crossrefs

A027748, A124010 are used in a formula defining this sequence.

Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).

Cf. A060175, A141810, A213925.

Programs

Haskell

a141809 n k = a141809_row n !! (k-1)
a141809_row 1 = [1]
a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
a141809_tabf = map a141809_row [1..]
-- Reinhard Zumkeller, Mar 18 2012

Mathematica

f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* Geoffrey Critzer, Apr 03 2015 *)

PARI

A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ M. F. Hasler, Oct 12 2018, updated Aug 19 2022

Formula

T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - Reinhard Zumkeller, Mar 15 2012