cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

2, 3, 5, 7, 9, 13, 21, 25, 37, 45, 57, 81, 133, 157, 193, 225, 253, 273, 325, 477, 613, 1821
Offset: 1

Views

Author

Gus Wiseman, Jan 04 2025

Keywords

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]]

A379841 Numbers that are the sum + product of some set of positive integers > 1. Positions of nonzeros in A379679.

Original entry on oeis.org

1, 4, 6, 8, 10, 11, 12, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
Offset: 1

Views

Author

Gus Wiseman, Jan 09 2025

Keywords

Examples

			For sum + product = 14 we have two possibilities: {7} or {2,4}; so 14 is in the sequence.

Crossrefs

The version allowing 1's is A326178.
Positions of nonzeros in A379679.
The complement is A379680.
The non-strict version is A379839, complement A379670.
For unique (instead of some) we have A379842.
Arrays counting multisets by sum and product: A379666, A379671, A379678.
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.
A379681 gives sum + product of prime indices.
Cf. A069016, A319000, A319057, A319916, A325036, A326152, A379720, A379840.

Programs

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

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jan 14 2025

Keywords

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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]

A379839 Numbers that are the sum + product of some multiset of positive integers > 1. Nonzeros of A379669.

Original entry on oeis.org

1, 4, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

Gus Wiseman, Jan 04 2025

Keywords

Comments

A superset of A379840.

Examples

			We have {2,5} with sum + product = 17, so 17 is in the sequence.
We have {2,3,5,5} with sum + product = 165, so 165 is in the sequence.

Crossrefs

The complement is A379670.
The strict version is A379841, see 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
A002865 counts partitions into parts > 1, strict A025147.
A318950 counts factorizations by sum.
Cf. A003963, A069016, A319000, A319057, A319916, A325036, A326152, A326178, A379680, A379720, A379840.

Programs

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

A379843 Least number x such that there are exactly n sets of positive integers > 1 with sum + product = x. Position of first appearance of n in A379679.

Original entry on oeis.org

2, 1, 14, 44, 47, 89, 119, 179, 159, 239, 335, 539, 599, 744, 359, 719, 839
Offset: 0

Views

Author

Gus Wiseman, Jan 15 2025

Keywords

Comments

Warning: Do not confuse with the multiset version A379543.

Examples

			We have a(4) = 47 due to the following four sets: {5,7}, {2,15}, {3,11}, {2,3,6}.

Crossrefs

For multisets instead of sets we have A379543, firsts of A379669.
Positions of first appearances in A379679, see 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.
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, A319057, A379670, A379680, A379720, A379839, A379840, A379841.

Programs

  • Mathematica
    nn=100;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[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[strfacs,n],Total[#]+Times@@#==n&]],{n,nn}];
Table[Position[s,k-1][[1,1]],{k,mnrm[s+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

Views

Author

Gus Wiseman, Jan 15 2025

Keywords

Comments

Warning: Do not confuse with the strict version A379843.

Examples

			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}.

Crossrefs

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.

Programs

  • Mathematica
    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]}]
Showing 1-6 of 6 results.

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