A331385 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) = n + k.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2, 0, 2, 0, 0, 0, 0, 0, 2, 6, 6, 4, 6, 2, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 4, 8, 6, 6, 7, 2, 4, 2, 0, 1, 0, 0, 0, 1

Offset

0,9

Links

Table of n, a(n) for n=0..87.

Example

Triangle begins:
  1
  0 1
  0 1 1
  0 0 2 1
  0 0 1 3 1
  0 0 0 2 3 1 1
  0 0 0 1 4 3 1 2
  0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  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 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 = 5 counts the following partitions:
  (11111)  (411)    (43)     (332)    (3222)   (22222)
           (3111)   (331)    (422)    (22221)
           (21111)  (421)    (3221)
                    (3211)   (22211)
                    (22111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Total[Prime/@#]==m&]], {n, 0, 10}, {m, n, Max@@Table[Total[Prime/@y], {y, IntegerPartitions[n]}]}]

Crossrefs

Row lengths are A331418.

Row sums are A000041.

Column sums are A331387.

Shifting row n to the right n times gives A331416.

Partitions 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, A330953, A330954, A331378, A331415.

Sequence in context: A108063 A164846 A363928 * A026729 A109466 A362763

Adjacent sequences: A331382 A331383 A331384 * A331386 A331387 A331388

Keyword

nonn,tabf

Author

Gus Wiseman, Jan 17 2020

Status

approved