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

%I A035306 #34 Jul 08 2025 21:32:33
%S A035306 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,
%T A035306 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,
%U A035306 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
%N A035306 List prime factors of each number in order (each prime factor is followed by its power). Start with 1 = {1,1}.
%C A035306 This entry also serves to show how to factor numbers in various languages.
%C A035306 Memo: in Maple, use ifactors, not ifactor!
%C A035306 Length of n-th row = 2*A001221(n). - _Reinhard Zumkeller_, Jan 10 2013
%H A035306 Reinhard Zumkeller, <a href="/A035306/b035306.txt">Rows n = 1..1000 of triangle, flattened</a>
%F A035306 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
%e A035306 The table starts as follows:
%e A035306    n  |  (p, valuation_p(n)) for primes p | n
%e A035306   ----+---------------------------------------
%e A035306    1  |  (1, 1),   (row 1, by definition of this sequence)
%e A035306    2  |  (2, 1),           (i.e.: 2 = 2^1)
%e A035306    3  |  (3, 1),
%e A035306    4  |  (2, 2),           (i.e.: 4 = 2^2)
%e A035306    5  |  (5, 1),
%e A035306    6  |  (2, 1), (3, 1),   (i.e.: 6 = 2^1 * 3^2)
%e A035306    7  |  (7, 1),
%e A035306    8  |  (2, 3),
%e A035306    9  |  (3, 2),
%e A035306    10 |  (2, 1), (5, 1),
%e A035306    11 |  (11, 1),
%e A035306    12 |  (2, 2), (3, 1),
%e A035306    13 |  (13, 1),
%e A035306    14 |  (2, 1), (7, 1),
%e A035306    15 |  (3, 1), (5, 1),
%e A035306    16 |  (2, 4),
%e A035306    17 |  (17, 1),
%e A035306    18 |  (2, 1), (3, 2),
%e A035306   ... | ...
%p A035306 ListTools[Flatten]([[[1, 1]], seq(op(2..-1, ifactors(n)), n=2..34)], 2); # _Peter Luschny_, Sep 02 2018
%t A035306 Flatten[ Array[ FactorInteger[ # ]&, 40 ] ]
%o A035306 (Haskell)
%o A035306 import Data.List (transpose)
%o A035306 a035306 n k = a035306_row n !! (k-1)
%o A035306 a035306_row 1 = [1,1]
%o A035306 a035306_row n = concat $ transpose [a027748_row n, a124010_row n]
%o A035306 a035306_tabf = map a035306_row [1..]
%o A035306 -- _Reinhard Zumkeller_, Jan 10 2013
%o A035306 (Magma) [ Factorization(n) : n in [1..120]];
%o A035306 (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
%o A035306 (PARI) A035306_row(n)=if(n>1, concat(Col(factor(n))~), [1, 1]) \\ _M. F. Hasler_, Jun 04 2024
%o A035306 (Python) A035306_row = lambda n: [x for f in factorint(n).items() for x in f]
%o A035306 from sympy import factorint # _M. F. Hasler_, Jun 06 2024
%Y A035306 Cf. A008474 (row sums, apart from initial row).
%Y A035306 Cf. A001221, A001222, A027748, A124010.
%K A035306 nonn,tabf
%O A035306 1,3
%A A035306 _N. J. A. Sloane_