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A381178 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.

%I A381178 #31 Mar 01 2025 12:19:05
%S A381178 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 A381178 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 A381178 3,2,5,1,1,2,13,3,3,1,2,2,7,1,29,1,1,1,2,3,5,1,31
%N A381178 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 A381178 Terms in each row are sorted; cf. A035306, where they are given in (base, exponent) groups.
%H A381178 Paolo Xausa, <a href="/A381178/b381178.txt">Table of n, a(n) for n = 2..11293</a> (rows 2..2500 of triangle, flattened).
%e A381178 Triangle begins:
%e A381178    [2]  1, 2;
%e A381178    [3]  1, 3;
%e A381178    [4]  2, 2;
%e A381178    [5]  1, 5;
%e A381178    [6]  1, 1, 2, 3;
%e A381178    [7]  1, 7;
%e A381178    [8]  2, 3;
%e A381178    [9]  2, 3;
%e A381178   [10]  1, 1, 2, 5;
%e A381178   ...
%e A381178 The prime factorization of 10 is 2^1*5^1 and the multiset of these bases and exponents is {1, 1, 2, 5}.
%e A381178 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 A381178 A381178row[n_] := Sort[Flatten[FactorInteger[n]]];
%t A381178 Array[A381178row, 30, 2]
%Y A381178 Cf. A000026 (row products), A001221 (row lengths, divided by 2), A008474 (row sums).
%Y A381178 Cf. A081812 (right border), A381212 (first column), A381576 (second column).
%Y A381178 Cf. A035306, A381203, A381204, A381398, A381401, A381403, A381404.
%K A381178 nonn,tabf,easy
%O A381178 2,2
%A A381178 _Paolo Xausa_, Feb 27 2025