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A331385
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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) = n + k.
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8
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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
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OFFSET
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0,9
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Total[Prime/@#]==m&]], {n, 0, 10}, {m, n, Max@@Table[Total[Prime/@y], {y, IntegerPartitions[n]}]}]
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CROSSREFS
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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.
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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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