%I #72 Sep 03 2023 09:36:00
%S 1,3,1,4,3,1,1,7,4,3,3,1,1,1,6,7,4,4,3,3,3,1,1,1,1,1,12,6,7,7,4,4,4,3,
%T 3,3,3,3,1,1,1,1,1,1,1,8,12,6,6,7,7,7,4,4,4,4,4,3,3,3,3,3,3,3,1,1,1,1,
%U 1,1,1,1,1,1,1,15,8,12,12,6,6,6,7,7,7,7,7,4,4,4,4,4,4,4
%N Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000203, the sum of divisors function.
%C Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.
%H Paolo Xausa, <a href="/A337209/b337209.txt">Table of n, a(n) for n = 1..10980</a> (rows 1..21 of the triangle, flattened)
%F T(n,k) = A000203(A176206(n,k)).
%e Triangle begins:
%e 1;
%e 3, 1;
%e 4, 3, 1, 1;
%e 7, 4, 3, 3, 1, 1, 1;
%e 6, 7, 4, 4, 3, 3, 3, 1, 1, 1, 1, 1;
%e 12, 6, 7, 7, 4, 4, 4, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
%e 8, 12, 6, 6, 7, 7, 7, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, ...
%e ...
%t A337209row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A337209row,10] (* _Paolo Xausa_, Sep 02 2023 *)
%o (PARI) f(n) = sum(k=0, n-1, numbpart(k));
%o T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--;); sigma(1+s);}
%o tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); );} \\ _Michel Marcus_, Jan 13 2021
%Y Sum of divisors of terms of A176206.
%Y Cf. A339278 (another version).
%Y Cf. A000070, A000203, A066186, A221529, A238442, A339258.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Nov 27 2020