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
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),
... | ...
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) 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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved