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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.
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%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