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%I #5 Jan 17 2020 17:41:50
%S 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,
%T 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,
%U 1,0,0,0,0,0,0,0,0,0,0,0,0,1,4,6,3,4,2
%N 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.
%e Triangle begins:
%e 1
%e 0 0 1
%e 0 0 0 1 1
%e 0 0 0 0 0 2 1
%e 0 0 0 0 0 0 1 3 1
%e 0 0 0 0 0 0 0 0 2 3 1 1
%e 0 0 0 0 0 0 0 0 0 1 4 3 1 2
%e 0 0 0 0 0 0 0 0 0 0 0 2 5 3 2 2 0 1
%e 0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 3 4 2 0 2
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
%e 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
%e 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
%e Row n = 8 counts the following partitions (empty column not shown):
%e (2222) (332) (44) (41111) (53) (611) (8)
%e (422) (431) (311111) (62) (5111) (71)
%e (3221) (3311) (2111111) (521)
%e (22211) (4211) (11111111)
%e (32111)
%e (221111)
%e Column k = 19 counts the following partitions:
%e (8) (6111) (532) (443) (33222)
%e (71) (51111) (622) (4331) (42222)
%e (5221) (4421) (322221)
%e (4111111) (33311) (2222211)
%e (31111111) (43211)
%e (211111111) (332111)
%e (422111)
%e (3221111)
%e (22211111)
%t maxm[n_]:=Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}];
%t Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==k&]],{n,0,10},{k,0,maxm[n]}]
%Y Row lengths are A331417.
%Y Row sums are A000041.
%Y Column sums are A000607.
%Y Shifting row n to the left n times gives A331385.
%Y Partitions whose Heinz number is divisible by their sum of primes: A330953.
%Y Partitions of whose sum of primes is divisible by their sum are A331379.
%Y Partitions whose product divides their sum of primes are A331381.
%Y Partitions whose product equals their sum of primes are A331383.
%Y Cf. A000040, A001414, A014689, A056239, A330950, A330954, A331378, A331387, A331415, A331418.
%K nonn,tabf
%O 0,15
%A _Gus Wiseman_, Jan 17 2020
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