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

1,3

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

Links

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

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

Example

1 = {1,1}, 2 = {2,1}, 3 = {3,1}, 4 = {2,2}, 5 = {5,1}, 6 = {2,1,3,1}, ...

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 ] ]

Prog

(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]];
(PARI) for (n=2, 256, print(factor(n))) [There has to be a better PARI code than this]
(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

Crossrefs

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

Cf. A001221, A001222, A027748, A124010.

Sequence in context: A029331 A037269 A124579 * A101691 A349445 A205379

Adjacent sequences: A035303 A035304 A035305 * A035307 A035308 A035309

Keyword

nonn,tabf

Author

N. J. A. Sloane.

Status

approved