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Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.
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%I #7 Nov 07 2023 08:24:58

%S 1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,1,1,0,0,1,1,0,

%T 0,1,0,1,1,1,1,1,2,1,0,1,1,0,0,1,0,1,1,1,1,1,2,2,1,1,1,0,0,1,1,0,0,1,

%U 0,1,1,1,1,1,2,2,2,3,2,0,2,1,0,1,1,0,0,1,0,1,1,1,1,1,2,2,2,3,3,2,2,2,2,1,1,0,0,1

%N Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

%C To illustrate the definition, the sum of primes indexed by all parts greater than one of the partition (5,2,2,1) is prime(5) + prime(2) + prime(2) = 17.

%e Triangle begins:

%e 1

%e 1

%e 1 0 0 1

%e 1 0 0 1 0 1

%e 1 0 0 1 0 1 1 1

%e 1 0 0 1 0 1 1 1 1 0 0 1

%e 1 0 0 1 0 1 1 1 1 1 2 1 0 1

%e 1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1

%e 1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1

%e 1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 2 2 2 2 1 1 0 0 1

%e 1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 4 2 3 2 0 3 1 0 0 0 0 0 1

%e 1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 5 4 4 3 3 3 2 3 0 1 0 0 1 0 1

%e The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).

%e The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).

%t Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#,OddQ]]==k&]], {n,0,10}, {k,0,If[n<=1,0,Prime[n]]}]

%Y Row lengths are A055670.

%Y Columns appear to converge to A099773.

%Y A bisected even version is A116598 (counts partitions by number of 1's).

%Y Counting all parts (not just > 1) gives A331416, shifted A331385.

%Y A000041 counts integer partitions, strict A000009 (also into odds).

%Y A113685 counts partitions by sum of odd parts, rank statistic A366528.

%Y A330953 counts partitions with Heinz number divisible by sum of primes.

%Y A331381 counts partitions with (product)|(sum of primes), equality A331383.

%Y Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.

%K nonn,tabf

%O 0,43

%A _Gus Wiseman_, Nov 01 2023