A379840 Numbers that are the sum + product of a unique multiset of positive integers > 1.

1, 4, 6, 10, 11, 12, 15, 16, 17, 18, 22, 27, 28, 30, 31, 43, 52, 58, 61, 67, 70, 73, 91, 97, 100, 102, 108, 115, 130, 145, 147, 148, 162, 165, 171, 217, 262, 277, 283, 291, 361, 430, 481, 508, 577, 633, 652, 682, 763, 1093, 1137, 1201, 1513, 1705, 2257, 2401, 2653, 3133, 4123, 5113, 5905

Offset: 1

Author

Gus Wiseman, Jan 08 2025

Keywords

• nonn

Examples

			The only multiset with no 1's and sum + product = 165 is {2,3,5,5}, so 165 is in the sequence.
		

Crossrefs

Positions of 1 in A379669 (zeros A379670).

For sets instead of multisets we have A379842, see A379841.

Arrays counting multisets by sum and product:

- partitions: A379666, antidiagonal sums A379667

- partitions without ones: A379668, antidiagonal sums A379669

- strict partitions: A379671, antidiagonal sums A379672

- strict partitions without ones: A379678, antidiagonal sums A379679

Counting and ranking multisets by comparing sum and product:

- same: A001055 (strict A045778), ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A326156, A326172, A379733

- greater: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721

- different: A379736, ranks A379722, see A111133

A002865 counts partitions into parts > 1, strict A025147.

A318950 counts factorizations by sum.

Cf. A003963, A069016, A319000, A319057, A319916, A325036, A326152, A379680.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Length[Select[Join@@Array[facs,n], Total[#]+Times@@#==n&]],{n,100}],1]

Extensions

More terms from Jinyuan Wang, Jan 12 2025