A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

1, 2, 2, 1, 3, 2, 1, 2, 2, 2, 1, 1, 4, 3, 2, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 4, 4, 2, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

T(n,k) is also the number of divisors of A336811(n,k).

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of the triangle, flattened)

Formula

a(m) = A000005(A336811(m)).

T(n,k) = A000005(A336811(n,k).

Example

Triangle begins:
  1;
  2;
  2, 1;
  3, 2, 1;
  2, 2, 2, 1, 1;
  4, 3, 2, 2, 2, 1, 1;
  2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1;
  4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1;
  3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  ...

Mathematica

A339304row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0, n-m], PartitionsP[m]-PartitionsP[m-1]], {m, 0, n-1}]]; Array[A339304row, 10] (* Paolo Xausa, Sep 01 2023 *)

Crossrefs

Number of divisors of A336811.

Row n has length A000041(n-1).

Every column gives A000005.

Row sums give A138137 (conjectured).

Cf. A135010, A138121, A138879, A187219.

Sequence in context: A281013 A190683 A181810 * A237578 A026146 A325519

Adjacent sequences: A339301 A339302 A339303 * A339305 A339306 A339307

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 29 2020

Status

approved