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%I #6 Jan 03 2025 23:28:40
%S 0,1,0,0,1,0,1,0,2,0,1,1,1,0,3,1,1,1,1,2,2,0,1,2,4,0,3,1,1,3,1,1,2,2,
%T 3,3,2,0,2,3,2,2,4,1,4,0,3,4,2,2,2,3,1,2,4,2,3,0,1,8,3,1,4,2,3,3,2,1,
%U 3,5,1,4,3,1,4,2,7,2,3,4,3,0,2,4,6,2,4,4
%N Number of finite multisets of positive integers > 1 with sum + product = n.
%e The partition (3,2,2) has sum + product equal to 7 + 12 = 19, so is counted under a(19).
%e The a(n) partitions for n = 4, 8, 14, 24, 59:
%e (2) (4) (7) (12) (9,5)
%e (2,2) (4,2) (4,4) (11,4)
%e (2,2,2) (4,2,2) (14,3)
%e (2,2,2,2) (19,2)
%e (4,4,3)
%e (11,2,2)
%e (4,3,2,2)
%e (3,2,2,2,2)
%t Table[Length[Select[Select[Join@@Array[IntegerPartitions,n+1,0],FreeQ[#,1]&],Total[#]+Times@@#==n&]],{n,0,30}]
%Y Arrays counting multisets by sum and product:
%Y - partitions: A379666, antidiagonal sums A379667
%Y - partitions without ones: A379668, antidiagonal sums A379669 (this) (zeros A379670)
%Y - strict partitions: A379671, antidiagonal sums A379672
%Y - strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
%Y Counting and ranking multisets by comparing sum and product:
%Y - same: A001055 (strict A045778), ranks A301987
%Y - divisible: A057567, ranks A326155
%Y - divisor: A057568, ranks A326149, see A326156, A326172, A379733
%Y - greater: A096276 shifted right, ranks A325038
%Y - greater or equal: A096276, ranks A325044
%Y - less: A114324, ranks A325037, see A318029
%Y - less or equal: A319005, ranks A379721
%Y - different: A379736, ranks A379722, see A111133
%Y A000041 counts integer partitions, strict A000009.
%Y A025147 counts strict partitions into parts > 1, non-strict A002865.
%Y A318950 counts factorizations by sum.
%Y A326622 counts factorizations with integer mean, strict A328966.
%Y Cf. A003963, A069016, A319000, A319057, A319916, A325036, A326152, A326178, A379720.
%K nonn
%O 0,9
%A _Gus Wiseman_, Jan 03 2025