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A331416
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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.
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18
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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
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OFFSET
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0,15
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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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]}]
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CROSSREFS
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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.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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