A379843 -id:A379843 - cp's OEIS frontend

A379842 Numbers that are the sum + product of a unique set of positive integers > 1. Positions of 1 in A379679.

1, 4, 6, 8, 10, 11, 12, 16, 17, 18, 19, 22, 24, 27, 28, 30, 31, 33, 36, 42, 43, 46, 48, 49, 52, 58, 61, 63, 66, 67, 70, 73, 85, 88, 91, 97, 100, 102, 105, 108, 115, 126, 130, 141, 145, 147, 148, 162, 171, 178, 192, 205, 211, 213, 226, 262, 277, 283, 288, 291

Offset: 1

Gus Wiseman, Jan 14 2025

nonn

			For sum + product = 29 we have two possibilities: {2,9} and {4,5}, so 29 is not in the sequence.
For sum + product = 33 we have only {2,3,4}, so 33 is in the sequence.

Positions of 1 in A379679, see A379843.
For at least one multiset we have A379839, complement A379670.
For multisets instead of sets we have A379840.
For at least one (instead of exactly one) we have A379841, complement A379680.
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.
A000041 counts integer partitions, strict A000009.
A001055 counts integer factorizations, strict A045778.
A002865 counts partitions into parts > 1, strict A025147.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A069016, A111133, A319000, A319057, A326152, A326178, A379720.

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

A379543 Least number x such that there are exactly n multisets of positive integers > 1 with sum + product = x. Position of first appearance of n in A379669.

Original entry on oeis.org

2, 1, 8, 14, 24, 69, 84, 76, 59, 179, 195, 159, 314, 449, 384, 984, 467, 359, 909, 744, 839

Offset: 0

Gus Wiseman, Jan 15 2025

Warning: Do not confuse with the strict version A379843.

			We have a(5) = 69 due to the following five multisets: {4,13}, {6,9}, {2,2,13}, {2,4,7}, {2,2,2,7}.

Positions of first appearances in A379669.
For sets instead of multisets we have A379843, firsts of A379679.
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.
A000041 counts integer partitions, strict A000009.
A001055 counts integer factorizations, strict A045778.
A002865 counts partitions into parts > 1, strict A025147.
A318950 counts factorizations by sum.
Cf. A069016, A111133, A319000, A379670, A379680, A379720, A379839, A379840, A379841, A379842.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
s=Table[Length[Select[Join@@Array[facs,n],Total[#]+Times@@#==n&]],{n,100}];
Table[Position[s,k-1][[1,1]],{k,mnrm[s+1]}]

cp's OEIS Frontend

A379842 Numbers that are the sum + product of a unique set of positive integers > 1. Positions of 1 in A379679.

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