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.

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A379671 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers with sum n and product k.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Jan 01 2025

Keywords

Comments

Counts finite sets of positive integers by sum and product.

Examples

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   1   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   1   1   0   0   0   0   0   0   0   0
   n=5:  0   0   0   1   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   1   2   0   1   0   0   0   0
   n=7:  0   0   0   0   0   1   1   1   0   1   0   1
   n=8:  0   0   0   0   0   0   1   1   0   1   0   2
   n=9:  0   0   0   0   0   0   0   1   1   0   0   1
  n=10:  0   0   0   0   0   0   0   0   1   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   1   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   1   1
The A(8,12) = 2 sets are: {2,6}, {1,3,4}.
The A(14,40) = 2 sets are: {4,10}, {1,5,8}.
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 1 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 1 0 0
   n+k=8: 0 0 0 0 1 0 0 0
   n+k=9: 0 0 0 0 0 1 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 1 0 0 0 0
  n+k=12: 0 0 0 0 0 0 2 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=11 counts the following sets:
  n=5: {2,3}
  n=6: {1,5}
so the 11th antidiagonal is: (0,0,0,0,0,1,1,0,0,0,0).

Crossrefs

Row sums are A000009 = strict partitions, non-strict A000041.
Column sums are 2*A045778 where A045778 = strict factorizations, non-strict A001055.
Antidiagonal sums are A379672, non-strict A379667 (zeros A379670).
Without ones we have A379678, antidiagonal sums A379679 (zeros A379680).
The non-strict version is A379666, without ones A379668.
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A002865, A003963, A028422, A069016, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

Programs

  • Mathematica
    nn=12;
tt=Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.

Original entry on oeis.org

4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 168, 178, 194, 202, 206, 214, 216, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 400, 422, 446, 454, 458, 466
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

Examples

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
  106: {1,16}
  118: {1,17}
  122: {1,18}

Crossrefs

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325044.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==Total[primeMS[#]]-1&]

Formula

A003963(a(n)) = A056239(a(n)) - 1.
a(n) = 2 * A301987(n).

A379668 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n into parts > 1 with product k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Dec 31 2024

Keywords

Comments

This table counts finite multisets of positive integers > 1 by sum and product. Compare to the triangle A318950.

Examples

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  0   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   0   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   0   2   0   0   0   0   0   0   0   0
   n=5:  0   0   0   0   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   0   1   0   2   1   0   0   0
   n=7:  0   0   0   0   0   0   1   0   0   1   0   2
   n=8:  0   0   0   0   0   0   0   1   0   0   0   1
   n=9:  0   0   0   0   0   0   0   0   1   0   0   0
  n=10:  0   0   0   0   0   0   0   0   0   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   0   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   0   1
For example, the A(11,48) = 3 partitions are: (4,4,3), (4,3,2,2), (3,2,2,2,2).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 0
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 0 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 0 0 0
   n+k=8: 0 0 0 0 2 0 0 0
   n+k=9: 0 0 0 0 0 0 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 0 0 0 0 0
  n+k=12: 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 0 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 2 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=14 counts the following partitions:
  n=6: (42), (222)
  n=7: (7)
so the 14th antidiagonal is: (0,0,0,0,0,0,2,1,0,0,0,0,0,0,0).

Crossrefs

Column sums are A001055 = factorizations, strict A045778.
Row sums are A002865 = partitions into parts > 1.
Take transpose and remove upper half (all zeros) to get A318950.
Allowing one gives A379666, antidiagonal sums A379667.
Antidiagonal sums are A379669, zeros A379670.
The strict case allowing ones is A379671, antidiagonal sums A379672.
The strict case is A379678, antidiagonal sums A379679 (zeros A379680).
A000041 counts integer partitions, strict A000009.
A316439 counts factorizations by length, A008284 partitions.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A325042, A326152, A326178, A379720.

Programs

  • Mathematica
    nn=15;
tt=Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

Formula

For n <= k we have A(n,k) = A318950(k,n).

A325039 Number of integer partitions of n with the same product of parts as their conjugate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 6, 2, 2, 4, 3, 5, 7, 6, 5, 7, 9, 10, 11, 18, 16, 19, 19, 16, 20, 20, 28, 39, 28, 40, 53, 45, 52, 59, 71, 61, 73, 97, 102, 95, 112, 131, 137, 148, 140, 166, 199, 181, 238, 251, 255, 289, 339, 344, 381, 398, 422, 464, 541, 555, 628, 677, 732
Offset: 0

Views

Author

Gus Wiseman, Mar 25 2019

Keywords

Comments

For example, the partition (6,4,1) with product 24 has conjugate (3,2,2,2,1,1) with product also 24.
The Heinz numbers of these partitions are given by A325040.

Examples

			The a(8) = 6 through a(15) = 6 integer partitions:
  (44)    (333)    (4321)   (641)     (4422)    (4432)     (6431)
  (332)   (51111)  (52111)  (4331)    (53211)   (6421)     (8411)
  (431)                     (322211)  (621111)  (53311)    (54221)
  (2222)                    (611111)            (432211)   (433211)
  (3221)                                        (7111111)  (632111)
  (4211)                                                   (7211111)
                                                           (42221111)

Crossrefs

Cf. A001055, A064573, A122111, A296150, A318950, A319000, A320322, A321649.
Cf. A325040, A325041, A325042, A325045.

Programs

  • Mathematica
    conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@#==Times@@conj[#]&]],{n,0,30}]

Extensions

More terms from Jinyuan Wang, Jun 27 2020

A379667 Number of finite multisets of positive integers with sum + product = n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 13, 14, 15, 17, 19, 19, 20, 22, 26, 26, 29, 30, 31, 34, 35, 36, 38, 40, 43, 46, 48, 48, 50, 53, 55, 57, 61, 62, 66, 66, 69, 73, 75, 77, 79, 82, 83, 85, 89, 91, 94, 94, 95, 103, 106, 107, 111, 113, 116, 119, 121
Offset: 0

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Examples

			The partition (2,2,1) has sum + product equal to 5 + 4 = 9, so is counted under a(9).
The a(0) = 0 through a(8) = 5 partitions:
  .  ()  (1)  (11)  (2)    (21)    (3)      (31)      (4)
                    (111)  (1111)  (211)    (2111)    (22)
                                   (11111)  (111111)  (311)
                                                      (21111)
                                                      (1111111)

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667 (this)
- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
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.
A025147 counts partitions into distinct parts > 1, non-strict A002865.
A316439 counts factorizations by length, partitions A008284.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A326152, A326178, A379720.

Programs

  • Mathematica
    Table[Length[Select[Join@@Array[IntegerPartitions,n+1,0],Total[#]+Times@@#==n&]],{n,0,30}]

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]]

A379672 Number of finite sets of positive integers with sum + product = n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 3, 1, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 3, 1, 2, 4, 4, 4, 3, 2, 4, 3, 1, 5, 5, 2, 3, 4, 3, 3, 5, 5, 4, 2, 1, 5, 6, 3, 4, 4, 3, 4, 3, 2, 4, 6, 4, 5, 6, 3, 4, 5, 4, 4, 4, 5, 5, 2, 2, 6, 7, 4, 3, 5
Offset: 0

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Comments

Antidiagonal sums of A379671, starting with 0.
The only zeros are a(0) and a(3).

Examples

			The a(n) sets for n = 2, 11, 20, 35, 47, 60:
  {1}  {1,5}  {10}     {3,8}    {5,7}    {30}
       {2,3}  {2,6}    {1,17}   {1,23}   {1,5,9}
              {1,3,4}  {2,11}   {2,15}   {2,4,6}
                       {1,4,6}  {3,11}   {1,2,19}
                                {2,3,6}  {1,3,14}
                                         {1,4,11}

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672 (this)
- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
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.
A025147 counts strict partitions into parts > 1, non-strict A002865.
A318950 counts factorizations by sum.
Cf. A003963, A028422, A096765, A319000, A319057, A319916, A325036, A326152, A326178.

Programs

  • Mathematica
    Table[Length[Select[Join@@Array[IntegerPartitions,n,0],UnsameQ@@#&&Total[#]+Times@@#==n&]],{n,0,30}]

Extensions

More terms from Jinyuan Wang, Jan 11 2025

A379679 Number of finite sets of positive integers > 1 with sum + product = n.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 0, 2, 2, 2, 2, 1, 1, 3, 0, 1, 4, 1, 1, 2, 2, 1, 2, 3, 2, 2, 0, 1, 4, 2, 1, 3, 1, 2, 2, 1, 1, 3, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 2, 0, 2, 4, 3, 1, 2, 3
Offset: 1

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Comments

Antidiagonal sums of A379678.

Examples

			The set {2,3,4,6} has sum 15 and product 144 so is counted under a(159).
The a(n) sets for n = 47, 89, 119, 159, 179, 239:
  {5,7}    {8,9}     {2,39}  {3,39}     {2,59}   {2,79}
  {2,15}   {2,29}    {3,29}  {4,31}     {3,44}   {3,59}
  {3,11}   {4,17}    {4,23}  {7,19}     {4,35}   {4,47}
  {2,3,6}  {5,14}    {5,19}  {9,15}     {5,29}   {5,39}
           {2,3,12}  {7,14}  {2,3,22}   {8,19}   {7,29}
                     {9,11}  {2,4,17}   {9,17}   {9,23}
                             {2,7,10}   {11,14}  {11,19}
                             {2,3,4,6}           {14,15}
                                                 {2,9,12}

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679 (this) (zeros A379680)
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.
A316439 counts factorizations by length, partitions A008284.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A003963, A069016, A319000, A319057, A319916, A326152, A326178, A379720.

Programs

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

A379669 Number of finite multisets of positive integers > 1 with sum + product = n.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 2, 2, 0, 1, 2, 4, 0, 3, 1, 1, 3, 1, 1, 2, 2, 3, 3, 2, 0, 2, 3, 2, 2, 4, 1, 4, 0, 3, 4, 2, 2, 2, 3, 1, 2, 4, 2, 3, 0, 1, 8, 3, 1, 4, 2, 3, 3, 2, 1, 3, 5, 1, 4, 3, 1, 4, 2, 7, 2, 3, 4, 3, 0, 2, 4, 6, 2, 4, 4
Offset: 0

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Author

Gus Wiseman, Jan 03 2025

Keywords

Examples

			The partition (3,2,2) has sum + product equal to 7 + 12 = 19, so is counted under a(19).
The a(n) partitions for n = 4, 8, 14, 24, 59:
  (2)  (4)    (7)      (12)       (9,5)
       (2,2)  (4,2)    (4,4)      (11,4)
              (2,2,2)  (4,2,2)    (14,3)
                       (2,2,2,2)  (19,2)
                                  (4,4,3)
                                  (11,2,2)
                                  (4,3,2,2)
                                  (3,2,2,2,2)

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669 (this) (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
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.
A025147 counts strict partitions into parts > 1, non-strict A002865.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A003963, A069016, A319000, A319057, A319916, A325036, A326152, A326178, A379720.

Programs

  • Mathematica
    Table[Length[Select[Select[Join@@Array[IntegerPartitions,n+1,0],FreeQ[#,1]&],Total[#]+Times@@#==n&]],{n,0,30}]

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jan 04 2025

Keywords

Comments

Is this sequence infinite?
Are all terms odd except for 2?

Examples

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

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.

Programs

  • 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]]
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