cp's OEIS Frontend

A379669 Number of finite multisets of positive integers > 1 with sum + product = n.

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