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.

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, 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, 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

Offset: 1

Author

Omar E. Pol, Nov 29 2020

Keywords

• nonn
• tabf

Comments

Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.

Examples

			Triangle begins:
  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, 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, 1, ...
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

Crossrefs

Row sums give A006128 (conjectured).

Cf. A000005, A000041, A176206, A221530, A221531, A337209.

Programs

Mathematica

A339258row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A339258row,10] (* Paolo Xausa, Sep 02 2023 *)

PARI

f(n) = sum(k=0, n-1, numbpart(k));
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);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

Formula

T(n,k) = A000005(A176206(n,k)).