A379679 - OEIS

%I #9 Jan 09 2025 08:01:40

%S 1,0,0,1,0,1,0,1,0,1,1,1,0,2,0,1,1,1,1,2,0,1,2,1,0,2,1,1,2,1,1,2,1,2,

%T 2,1,0,2,2,2,2,1,1,3,0,1,4,1,1,2,2,1,2,3,2,2,0,1,4,2,1,3,1,2,2,1,1,3,

%U 3,1,4,2,1,3,2,2,2,2,3,2,0,2,4,3,1,2,3

%N Number of finite sets of positive integers > 1 with sum + product = n.

%C Antidiagonal sums of A379678.

%e The set {2,3,4,6} has sum 15 and product 144 so is counted under a(159).

%e The a(n) sets for n = 47, 89, 119, 159, 179, 239:

%e   {5,7}    {8,9}     {2,39}  {3,39}     {2,59}   {2,79}

%e   {2,15}   {2,29}    {3,29}  {4,31}     {3,44}   {3,59}

%e   {3,11}   {4,17}    {4,23}  {7,19}     {4,35}   {4,47}

%e   {2,3,6}  {5,14}    {5,19}  {9,15}     {5,29}   {5,39}

%e            {2,3,12}  {7,14}  {2,3,22}   {8,19}   {7,29}

%e                      {9,11}  {2,4,17}   {9,17}   {9,23}

%e                              {2,7,10}   {11,14}  {11,19}

%e                              {2,3,4,6}           {14,15}

%e                                                  {2,9,12}

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[Join@@Array[facs,n],UnsameQ@@#&&Total[#]+Times@@#==n&]],{n,100}]

%Y Arrays counting multisets by sum and product:

%Y - partitions: A379666, antidiagonal sums A379667

%Y - partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)

%Y - strict partitions: A379671, antidiagonal sums A379672

%Y - strict partitions without ones: A379678, antidiagonal sums A379679 (this) (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 A002865 counts partitions into parts > 1, strict A025147.

%Y A316439 counts factorizations by length, partitions A008284.

%Y A318950 counts factorizations by sum.

%Y A326622 counts factorizations with integer mean, strict A328966.

%Y Cf. A003963, A069016, A319000, A319057, A319916, A326152, A326178, A379720.

%K nonn

%O 1,14

%A _Gus Wiseman_, Jan 03 2025