A381178 - OEIS

%I #31 Mar 01 2025 12:19:05

%S 1,2,1,3,2,2,1,5,1,1,2,3,1,7,2,3,2,3,1,1,2,5,1,11,1,2,2,3,1,13,1,1,2,

%T 7,1,1,3,5,2,4,1,17,1,2,2,3,1,19,1,2,2,5,1,1,3,7,1,1,2,11,1,23,1,2,3,

%U 3,2,5,1,1,2,13,3,3,1,2,2,7,1,29,1,1,1,2,3,5,1,31

%N Irregular triangle read by rows, where row n lists the elements of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n.

%C Terms in each row are sorted; cf. A035306, where they are given in (base, exponent) groups.

%H Paolo Xausa, <a href="/A381178/b381178.txt">Table of n, a(n) for n = 2..11293</a> (rows 2..2500 of triangle, flattened).

%e Triangle begins:

%e    [2]  1, 2;

%e    [3]  1, 3;

%e    [4]  2, 2;

%e    [5]  1, 5;

%e    [6]  1, 1, 2, 3;

%e    [7]  1, 7;

%e    [8]  2, 3;

%e    [9]  2, 3;

%e   [10]  1, 1, 2, 5;

%e   ...

%e The prime factorization of 10 is 2^1*5^1 and the multiset of these bases and exponents is {1, 1, 2, 5}.

%e The prime factorization of 132 is 2^2*3^1*11^1 and the multiset of these bases and exponents is {1, 1, 2, 2, 3, 11}.

%t A381178row[n_] := Sort[Flatten[FactorInteger[n]]];

%t Array[A381178row, 30, 2]

%Y Cf. A000026 (row products), A001221 (row lengths, divided by 2), A008474 (row sums).

%Y Cf. A081812 (right border), A381212 (first column), A381576 (second column).

%Y Cf. A035306, A381203, A381204, A381398, A381401, A381403, A381404.

%K nonn,tabf,easy,new

%O 2,2

%A _Paolo Xausa_, Feb 27 2025