A379670 Numbers that are not the sum + product of any multiset of positive integers > 1. Zeros of A379669.

2, 3, 5, 7, 9, 13, 21, 25, 37, 45, 57, 81, 133, 157, 193, 225, 253, 273, 325, 477, 613, 1821

Offset: 1

Author

Gus Wiseman, Jan 04 2025

Keywords

• nonn
• more

Comments

Is this sequence infinite?

Are all terms odd except for 2?

Examples

			The partition (3,2,2) has sum + product equal to 7 + 12 = 19, so 19 is not in the sequence.
		

Crossrefs

The strict case is A379680.

The complement is A379839, a superset of A379840.

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

A000041 counts integer partitions, strict A000009.

A002865 counts partitions into parts > 1, strict A025147.

A318950 counts factorizations by sum.

Cf. A003963, A069016, A319000, A319057, A325036, A326152, A379720.

Programs

Mathematica

nn=1000;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Complement[Range[nn],Total[#]+Times@@#&/@Join@@Array[facs,nn]]