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A331416 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.

%I A331416 #5 Jan 17 2020 17:41:50
%S A331416 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 A331416 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 A331416 1,0,0,0,0,0,0,0,0,0,0,0,0,1,4,6,3,4,2
%N A331416 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 A331416 Triangle begins:
%e A331416   1
%e A331416   0 0 1
%e A331416   0 0 0 1 1
%e A331416   0 0 0 0 0 2 1
%e A331416   0 0 0 0 0 0 1 3 1
%e A331416   0 0 0 0 0 0 0 0 2 3 1 1
%e A331416   0 0 0 0 0 0 0 0 0 1 4 3 1 2
%e A331416   0 0 0 0 0 0 0 0 0 0 0 2 5 3 2 2 0 1
%e A331416   0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 3 4 2 0 2
%e A331416   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 A331416   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 A331416   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 A331416 Row n = 8 counts the following partitions (empty column not shown):
%e A331416   (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
%e A331416           (422)    (431)     (311111)   (62)        (5111)  (71)
%e A331416           (3221)   (3311)    (2111111)  (521)
%e A331416           (22211)  (4211)               (11111111)
%e A331416                    (32111)
%e A331416                    (221111)
%e A331416 Column k = 19 counts the following partitions:
%e A331416   (8)   (6111)   (532)        (443)       (33222)
%e A331416   (71)  (51111)  (622)        (4331)      (42222)
%e A331416                  (5221)       (4421)      (322221)
%e A331416                  (4111111)    (33311)     (2222211)
%e A331416                  (31111111)   (43211)
%e A331416                  (211111111)  (332111)
%e A331416                               (422111)
%e A331416                               (3221111)
%e A331416                               (22211111)
%t A331416 maxm[n_]:=Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}];
%t A331416 Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==k&]],{n,0,10},{k,0,maxm[n]}]
%Y A331416 Row lengths are A331417.
%Y A331416 Row sums are A000041.
%Y A331416 Column sums are A000607.
%Y A331416 Shifting row n to the left n times gives A331385.
%Y A331416 Partitions whose Heinz number is divisible by their sum of primes: A330953.
%Y A331416 Partitions of whose sum of primes is divisible by their sum are A331379.
%Y A331416 Partitions whose product divides their sum of primes are A331381.
%Y A331416 Partitions whose product equals their sum of primes are A331383.
%Y A331416 Cf. A000040, A001414, A014689, A056239, A330950, A330954, A331378, A331387, A331415, A331418.
%K A331416 nonn,tabf
%O A331416 0,15
%A A331416 _Gus Wiseman_, Jan 17 2020