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

1,1

Comments

Is this sequence infinite?

Are all terms odd except for 2?

All terms beyond 7 are of the form 2p-1 where p is prime. All even numbers beyond 2 can be obtained from {m} -> 2m. For odd numbers, if n+1 = ab with 2 < a,b, then {a-1,b-1} -> ab-1 = n. This can be achieved unless n+1 is 8 or twice a prime. - Martin Fuller, Dec 19 2025

Links

Table of n, a(n) for n=1..22.

Example

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

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]]

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.

Sequence in context: A240063 A216374 A240448 * A218614 A338360 A028378

Adjacent sequences: A379667 A379668 A379669 * A379671 A379672 A379673

Keyword

nonn,more

Author

Gus Wiseman, Jan 04 2025

Status

approved