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A337209 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.

%I A337209 #72 Sep 03 2023 09:36:00
%S A337209 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 A337209 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 A337209 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 A337209 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 A337209 Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.
%H A337209 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 A337209 T(n,k) = A000203(A176206(n,k)).
%e A337209 Triangle begins:
%e A337209    1;
%e A337209    3,  1;
%e A337209    4,  3, 1, 1;
%e A337209    7,  4, 3, 3, 1, 1, 1;
%e A337209    6,  7, 4, 4, 3, 3, 3, 1, 1, 1, 1, 1;
%e A337209   12,  6, 7, 7, 4, 4, 4, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
%e A337209    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 A337209   ...
%t A337209 A337209row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A337209row,10] (* _Paolo Xausa_, Sep 02 2023 *)
%o A337209 (PARI) f(n) = sum(k=0, n-1, numbpart(k));
%o A337209 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 A337209 tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); );} \\ _Michel Marcus_, Jan 13 2021
%Y A337209 Sum of divisors of terms of A176206.
%Y A337209 Cf. A339278 (another version).
%Y A337209 Cf. A000070, A000203, A066186, A221529, A238442, A339258.
%K A337209 nonn,tabf
%O A337209 1,2
%A A337209 _Omar E. Pol_, Nov 27 2020