A035306 List prime factors of each number in order (each prime factor is followed by its power). Start with 1 = {1,1}.

1, 1, 2, 1, 3, 1, 2, 2, 5, 1, 2, 1, 3, 1, 7, 1, 2, 3, 3, 2, 2, 1, 5, 1, 11, 1, 2, 2, 3, 1, 13, 1, 2, 1, 7, 1, 3, 1, 5, 1, 2, 4, 17, 1, 2, 1, 3, 2, 19, 1, 2, 2, 5, 1, 3, 1, 7, 1, 2, 1, 11, 1, 23, 1, 2, 3, 3, 1, 5, 2, 2, 1, 13, 1, 3, 3, 2, 2, 7, 1, 29, 1, 2, 1, 3, 1, 5, 1, 31, 1, 2, 5, 3, 1, 11, 1, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabf

Comments

This entry also serves to show how to factor numbers in various languages.

Memo: in Maple, use ifactors, not ifactor!

Length of n-th row = 2*A001221(n). - Reinhard Zumkeller, Jan 10 2013

Examples

			The table starts as follows:
   n  |  (p, valuation_p(n)) for primes p | n
  ----+---------------------------------------
   1  |  (1, 1),   (row 1, by definition of this sequence)
   2  |  (2, 1),           (i.e.: 2 = 2^1)
   3  |  (3, 1),
   4  |  (2, 2),           (i.e.: 4 = 2^2)
   5  |  (5, 1),
   6  |  (2, 1), (3, 1),   (i.e.: 6 = 2^1 * 3^2)
   7  |  (7, 1),
   8  |  (2, 3),
   9  |  (3, 2),
   10 |  (2, 1), (5, 1),
   11 |  (11, 1),
   12 |  (2, 2), (3, 1),
   13 |  (13, 1),
   14 |  (2, 1), (7, 1),
   15 |  (3, 1), (5, 1),
   16 |  (2, 4),
   17 |  (17, 1),
   18 |  (2, 1), (3, 2),
  ... | ...
		

Links

• Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Crossrefs

Cf. A008474 (row sums, apart from initial row).

Cf. A001221, A001222, A027748, A124010.

Programs

Haskell

import Data.List (transpose)
a035306 n k = a035306_row n !! (k-1)
a035306_row 1 = [1,1]
a035306_row n = concat $ transpose [a027748_row n, a124010_row n]
a035306_tabf = map a035306_row [1..]
-- Reinhard Zumkeller, Jan 10 2013

Magma

[ Factorization(n) : n in [1..120]];

Maple

ListTools[Flatten]([[[1, 1]], seq(op(2..-1, ifactors(n)), n=2..34)], 2); # Peter Luschny, Sep 02 2018

Mathematica

Flatten[ Array[ FactorInteger[ # ]&, 40 ] ]

PARI

upto(n) = {n = max(n, 1); my(res = List([1, 1])); for(i = 2, n, f = factor(i); for(j = 1, #f~, listput(res, f[j, 1]); listput(res, f[j, 2]))); res} \\ David A. Corneth, Sep 02 2018

PARI

A035306_row(n)=if(n>1, concat(Col(factor(n))~), [1, 1]) \\ M. F. Hasler, Jun 04 2024

Python

A035306_row = lambda n: [x for f in factorint(n).items() for x in f]
from sympy import factorint # M. F. Hasler, Jun 06 2024

Formula

For 1 <= k <= A001221(n): T(n,2*k-1) = A027748(n,k), T(n,2*k) = A124010(n,k). - Reinhard Zumkeller, Jan 10 2013