A331416 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = k.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2

Offset

0,15

Links

Table of n, a(n) for n=0..86.

Example

Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 0 0 2 1
  0 0 0 0 0 0 1 3 1
  0 0 0 0 0 0 0 0 2 3 1 1
  0 0 0 0 0 0 0 0 0 1 4 3 1 2
  0 0 0 0 0 0 0 0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
Row n = 8 counts the following partitions (empty column not shown):
  (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
          (422)    (431)     (311111)   (62)        (5111)  (71)
          (3221)   (3311)    (2111111)  (521)
          (22211)  (4211)               (11111111)
                   (32111)
                   (221111)
Column k = 19 counts the following partitions:
  (8)   (6111)   (532)        (443)       (33222)
  (71)  (51111)  (622)        (4331)      (42222)
                 (5221)       (4421)      (322221)
                 (4111111)    (33311)     (2222211)
                 (31111111)   (43211)
                 (211111111)  (332111)
                              (422111)
                              (3221111)
                              (22211111)

Mathematica

maxm[n_]:=Max@@Table[Total[Prime/@y], {y, IntegerPartitions[n]}];
Table[Length[Select[IntegerPartitions[n], Total[Prime/@#]==k&]], {n, 0, 10}, {k, 0, maxm[n]}]

Crossrefs

Row lengths are A331417.

Row sums are A000041.

Column sums are A000607.

Shifting row n to the left n times gives A331385.

Partitions whose Heinz number is divisible by their sum of primes: A330953.

Partitions of whose sum of primes is divisible by their sum are A331379.

Partitions whose product divides their sum of primes are A331381.

Partitions whose product equals their sum of primes are A331383.

Cf. A000040, A001414, A014689, A056239, A330950, A330954, A331378, A331387, A331415, A331418.

Sequence in context: A342465 A124749 A350890 * A127844 A017877 A295976

Adjacent sequences: A331413 A331414 A331415 * A331417 A331418 A331419

Keyword

nonn,tabf

Author

Gus Wiseman, Jan 17 2020

Status

approved