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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.
8
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
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.
Sequence in context: A108063 A164846 A363928 * A026729 A109466 A362763
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 17 2020
STATUS
approved