cp's OEIS Frontend

A366851 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.

%I A366851 #7 Nov 07 2023 08:24:58
%S A366851 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 A366851 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 A366851 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 A366851 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 A366851 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 A366851 Triangle begins:
%e A366851   1
%e A366851   1
%e A366851   1 0 0 1
%e A366851   1 0 0 1 0 1
%e A366851   1 0 0 1 0 1 1 1
%e A366851   1 0 0 1 0 1 1 1 1 0 0 1
%e A366851   1 0 0 1 0 1 1 1 1 1 2 1 0 1
%e A366851   1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1
%e A366851   1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1
%e A366851   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 A366851   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 A366851   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 A366851 The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).
%e A366851 The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).
%t A366851 Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#,OddQ]]==k&]], {n,0,10}, {k,0,If[n<=1,0,Prime[n]]}]
%Y A366851 Row lengths are A055670.
%Y A366851 Columns appear to converge to A099773.
%Y A366851 A bisected even version is A116598 (counts partitions by number of 1's).
%Y A366851 Counting all parts (not just > 1) gives A331416, shifted A331385.
%Y A366851 A000041 counts integer partitions, strict A000009 (also into odds).
%Y A366851 A113685 counts partitions by sum of odd parts, rank statistic A366528.
%Y A366851 A330953 counts partitions with Heinz number divisible by sum of primes.
%Y A366851 A331381 counts partitions with (product)|(sum of primes), equality A331383.
%Y A366851 Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.
%K A366851 nonn,tabf
%O A366851 0,43
%A A366851 _Gus Wiseman_, Nov 01 2023