A049822 -id:A049822 - cp's OEIS frontend

A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 4, 2, 2, 1, 1, 1, 2, 4, 2, 4, 2, 2, 1, 1, 1, 3, 4, 5, 3, 4, 2, 2, 1, 1, 1, 1, 3, 4, 5, 3, 4, 2, 2, 1, 1, 1, 5, 4, 6, 5, 6, 3, 4, 2, 2, 1, 1, 1, 1, 5, 4, 6, 5, 6, 3, 4, 2, 2, 1, 1, 1, 3, 4, 7, 6, 7, 6, 6, 3, 4, 2, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, Sep 20 2006

			Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 1, 2, 1, 1;
  1, 3, 2, 2, 1, 1;
  ...
T(6,3) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.

Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..50 from G. C. Greubel)
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5.

Column k=1..4 give A057427, A032741, A049822, A121895.

Row sums give A003238.

T[, 1] = 1; T[n, k_] := T[n, k] = DivisorSum[n, If[#==1, 0, T[#-1, k-1]]& ]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 30 2016 *)

T(n,1) = 1. T(n,k+1) = Sum_{d|n, d1} T(d-1,k).

A121895 Number of partitions of n into 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 2, 5, 4, 6, 4, 7, 5, 10, 5, 8, 6, 11, 8, 13, 6, 12, 7, 13, 9, 15, 8, 16, 10, 17, 10, 14, 10, 20, 11, 14, 10, 23, 10, 22, 12, 21, 15, 20, 8, 21, 12, 23, 18, 24, 11, 20, 15, 30, 18, 21, 8, 28, 14, 21, 18, 32, 16, 34, 16, 22, 15, 28, 14, 33, 14, 22, 20, 31, 18, 32, 15

Offset: 1

Zak Seidov, Sep 01 2006

nonn

			a(36)=20 because there are 20 partitions of 36 in 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d:
{33, 1, 1, 1}, {32, 2, 1, 1}, {30, 2, 2, 2}, {28, 4, 2, 2}, {27, 3, 3, 3}, {25, 5, 5, 1}, {24, 8, 2, 2}, {24, 6, 3, 3}, {24, 4, 4, 4}, {21, 7, 7, 1}, {20, 10, 5, 1}, {18, 6, 6, 6}, {17, 17, 1, 1}, {16, 16, 2, 2}, {16, 8, 8, 4}, {15, 15, 5, 1}, {15, 15, 3, 3}, {14, 14, 7, 1}, {12, 12, 6, 6}, {9, 9, 9, 9}.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A026810 = number of partitions of n into exactly 4 parts.
Column 4 of A122934.
Cf. A049822, A003238.

a(n) = Sum_{d|n, d>1} A122935(d-1). - Franklin T. Adams-Watters, Sep 20 2006

A086898 a(n) = Sum_{d|n} tau(d-1).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 5, 6, 7, 4, 9, 6, 7, 9, 9, 5, 11, 6, 11, 12, 9, 4, 13, 11, 10, 10, 13, 6, 17, 8, 11, 12, 10, 11, 19, 9, 9, 12, 17, 8, 19, 8, 13, 19, 11, 4, 19, 14, 18, 13, 16, 6, 17, 15, 19, 16, 11, 4, 25, 12, 11, 20, 17, 16, 23, 8, 14, 12, 21, 8, 25, 12, 12, 21, 17, 14, 22, 8, 23, 20

Offset: 1

Vladeta Jovovic, Aug 23 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000005, A049822.

a[n_] := DivisorSum[n, If[# == 1, 0, DivisorSigma[0, # - 1]] &]; Array[a, 100] (* G. C. Greubel, Dec 11 2017 *)

cp's OEIS Frontend

A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

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A121895 Number of partitions of n into 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d.

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Formula

A086898 a(n) = Sum_{d|n} tau(d-1).

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Author

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Mathematica