cp's OEIS Frontend

A379672 Number of finite sets of positive integers with sum + product = n.

%I A379672 #17 Jan 11 2025 10:26:25
%S A379672 0,1,1,0,1,1,1,1,1,1,1,2,2,1,2,2,1,2,2,2,3,2,1,3,3,1,2,3,2,3,3,2,3,3,
%T A379672 3,4,3,1,2,4,4,4,3,2,4,3,1,5,5,2,3,4,3,3,5,5,4,2,1,5,6,3,4,4,3,4,3,2,
%U A379672 4,6,4,5,6,3,4,5,4,4,4,5,5,2,2,6,7,4,3,5
%N A379672 Number of finite sets of positive integers with sum + product = n.
%C A379672 Antidiagonal sums of A379671, starting with 0.
%C A379672 The only zeros are a(0) and a(3).
%e A379672 The a(n) sets for n = 2, 11, 20, 35, 47, 60:
%e A379672   {1}  {1,5}  {10}     {3,8}    {5,7}    {30}
%e A379672        {2,3}  {2,6}    {1,17}   {1,23}   {1,5,9}
%e A379672               {1,3,4}  {2,11}   {2,15}   {2,4,6}
%e A379672                        {1,4,6}  {3,11}   {1,2,19}
%e A379672                                 {2,3,6}  {1,3,14}
%e A379672                                          {1,4,11}
%t A379672 Table[Length[Select[Join@@Array[IntegerPartitions,n,0],UnsameQ@@#&&Total[#]+Times@@#==n&]],{n,0,30}]
%Y A379672 Arrays counting multisets by sum and product:
%Y A379672 - partitions: A379666, antidiagonal sums A379667
%Y A379672 - partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
%Y A379672 - strict partitions: A379671, antidiagonal sums A379672 (this)
%Y A379672 - strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
%Y A379672 Counting and ranking multisets by comparing sum and product:
%Y A379672 - same: A001055 (strict A045778), ranks A301987
%Y A379672 - divisible: A057567, ranks A326155
%Y A379672 - divisor: A057568, ranks A326149, see A326156, A326172, A379733
%Y A379672 - greater: A096276 shifted right, ranks A325038
%Y A379672 - greater or equal: A096276, ranks A325044
%Y A379672 - less: A114324, ranks A325037, see A318029
%Y A379672 - less or equal: A319005, ranks A379721
%Y A379672 - different: A379736, ranks A379722, see A111133
%Y A379672 A000041 counts integer partitions, strict A000009.
%Y A379672 A025147 counts strict partitions into parts > 1, non-strict A002865.
%Y A379672 A318950 counts factorizations by sum.
%Y A379672 Cf. A003963, A028422, A096765, A319000, A319057, A319916, A325036, A326152, A326178.
%K A379672 nonn
%O A379672 0,12
%A A379672 _Gus Wiseman_, Jan 03 2025
%E A379672 More terms from _Jinyuan Wang_, Jan 11 2025