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A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

%I A339304 #33 Dec 22 2024 14:28:35
%S A339304 1,2,2,1,3,2,1,2,2,2,1,1,4,3,2,2,2,1,1,2,2,3,2,2,2,2,1,1,1,1,4,4,2,3,
%T A339304 3,2,2,2,2,2,2,1,1,1,1,3,2,4,2,2,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,4,
%U A339304 4,2,4,4,2,2,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1
%N A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.
%C A339304 T(n,k) is also the number of divisors of A336811(n,k).
%C A339304 Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.
%H A339304 Paolo Xausa, <a href="/A339304/b339304.txt">Table of n, a(n) for n = 1..11732</a> (rows 1..27 of the triangle, flattened)
%F A339304 a(m) = A000005(A336811(m)).
%F A339304 T(n,k) = A000005(A336811(n,k)).
%e A339304 Triangle begins:
%e A339304   1;
%e A339304   2;
%e A339304   2, 1;
%e A339304   3, 2, 1;
%e A339304   2, 2, 2, 1, 1;
%e A339304   4, 3, 2, 2, 2, 1, 1;
%e A339304   2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1;
%e A339304   4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1;
%e A339304   3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
%e A339304   ...
%t A339304 A339304row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]];Array[A339304row,10] (* _Paolo Xausa_, Sep 01 2023 *)
%Y A339304 Number of divisors of A336811.
%Y A339304 Row n has length A000041(n-1).
%Y A339304 Every column gives A000005.
%Y A339304 Row sums give A138137 (conjectured).
%Y A339304 Cf. A135010, A138121, A138879, A187219.
%K A339304 nonn,tabf
%O A339304 1,2
%A A339304 _Omar E. Pol_, Nov 29 2020