cp's OEIS Frontend

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.

%I A331385 #6 Jan 17 2020 17:41:25
%S A331385 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,
%T A331385 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,
%U A331385 0,0,0,0,0,1,4,8,6,6,7,2,4,2,0,1,0,0,0,1
%N 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.
%e A331385 Triangle begins:
%e A331385   1
%e A331385   0 1
%e A331385   0 1 1
%e A331385   0 0 2 1
%e A331385   0 0 1 3 1
%e A331385   0 0 0 2 3 1 1
%e A331385   0 0 0 1 4 3 1 2
%e A331385   0 0 0 0 2 5 3 2 2 0 1
%e A331385   0 0 0 0 1 4 6 3 4 2 0 2
%e A331385   0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
%e A331385   0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
%e A331385   0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
%e A331385 Row n = 8 counts the following partitions (empty column not shown):
%e A331385   (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
%e A331385           (422)    (431)     (311111)   (62)        (5111)  (71)
%e A331385           (3221)   (3311)    (2111111)  (521)
%e A331385           (22211)  (4211)               (11111111)
%e A331385                    (32111)
%e A331385                    (221111)
%e A331385 Column k = 5 counts the following partitions:
%e A331385   (11111)  (411)    (43)     (332)    (3222)   (22222)
%e A331385            (3111)   (331)    (422)    (22221)
%e A331385            (21111)  (421)    (3221)
%e A331385                     (3211)   (22211)
%e A331385                     (22111)
%t A331385 Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==m&]],{n,0,10},{m,n,Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]}]
%Y A331385 Row lengths are A331418.
%Y A331385 Row sums are A000041.
%Y A331385 Column sums are A331387.
%Y A331385 Shifting row n to the right n times gives A331416.
%Y A331385 Partitions whose sum of primes is divisible by their sum are A331379.
%Y A331385 Partitions whose product divides their sum of primes are A331381.
%Y A331385 Partitions whose product equals their sum of primes are A331383.
%Y A331385 Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415.
%K A331385 nonn,tabf
%O A331385 0,9
%A A331385 _Gus Wiseman_, Jan 17 2020