A379680 Numbers that are not the sum + product of any set of positive integers > 1. Zeros of A379679.

2, 3, 5, 7, 9, 13, 15, 21, 25, 37, 45, 57, 81, 93, 121, 133, 157, 165, 193, 217, 225, 253, 273, 297, 325, 477, 525, 613, 981, 1201, 1213, 1317, 1813, 1821, 2401, 4273, 5113, 5905, 7477

Offset

1,1

Comments

Is this sequence infinite?

Are all terms odd except for 2?

All terms beyond 15 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 16 or twice a prime. - Martin Fuller, Dec 19 2025

Links

Table of n, a(n) for n=1..39.

Example

The set {2,3,4} has sum + product equal to 9 + 24 = 33, so 33 is not in the sequence.

Mathematica

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

Crossrefs

The non-strict version is A379670.

The complement is A379841, a superset of A379842.

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.

A326622 counts factorizations with integer mean, strict A328966.

Cf. A003963, A069016, A319000, A319057, A325036, A326178, A379720.

Sequence in context: A394033 A263647 A028870 * A338356 A057886 A367630

Adjacent sequences: A379677 A379678 A379679 * A379681 A379682 A379683

Keyword

nonn,look,more

Author

Gus Wiseman, Jan 04 2025

Status

approved