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A339258 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.

%I A339258 #39 Sep 03 2023 09:35:54
%S A339258 1,2,1,2,2,1,1,3,2,2,2,1,1,1,2,3,2,2,2,2,2,1,1,1,1,1,4,2,3,3,2,2,2,2,
%T A339258 2,2,2,2,1,1,1,1,1,1,1,2,4,2,2,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,
%U A339258 1,1,1,1,1,1,1,4,2,4,4,2,2,2,3,3,3,3,3,2,2,2,2,2,2,2
%N A339258 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.
%C A339258 Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.
%H A339258 Paolo Xausa, <a href="/A339258/b339258.txt">Table of n, a(n) for n = 1..10980</a> (rows 1..21 of the triangle, flattened)
%F A339258 T(n,k) = A000005(A176206(n,k)).
%e A339258 Triangle begins:
%e A339258   1;
%e A339258   2, 1;
%e A339258   2, 2, 1, 1;
%e A339258   3, 2, 2, 2, 1, 1, 1;
%e A339258   2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1;
%e A339258   4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
%e A339258   2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, ...
%e A339258   ...
%t A339258 A339258row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A339258row,10] (* _Paolo Xausa_, Sep 02 2023 *)
%o A339258 (PARI) f(n) = sum(k=0, n-1, numbpart(k));
%o A339258 T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (numdiv(n))); my(s=0); while (k <= f(n-1), s++; n--;); numdiv(1+s);}
%o A339258 tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ _Michel Marcus_, Jan 13 2021
%Y A339258 Row sums give A006128 (conjectured).
%Y A339258 Cf. A000005, A000041, A176206, A221530, A221531, A337209.
%K A339258 nonn,tabf
%O A339258 1,2
%A A339258 _Omar E. Pol_, Nov 29 2020