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 1-7 of 7 results.

A367394 Number of integer partitions of n whose length is a semi-sum of the parts.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 6, 7, 14, 15, 25, 30, 46, 54, 80, 97, 139, 169, 229, 282, 382, 461, 607, 746, 962, 1173, 1499, 1817, 2302, 2787, 3467, 4201, 5216, 6260, 7702, 9261, 11294, 13524, 16418, 19572, 23658, 28141, 33756, 40081, 47949, 56662, 67493, 79639
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2023

Keywords

Comments

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Examples

			For the partition y = (3,3,2,1) we have 4 = 3 + 1, so y is counted under a(9).
The a(2) = 1 through a(10) = 14 partitions:
  (11)  .  (211)  (221)  (321)   (421)   (521)    (621)    (721)
                         (2211)  (2221)  (2222)   (3222)   (3322)
                         (3111)  (3211)  (3221)   (3321)   (3331)
                                         (3311)   (4221)   (4222)
                                         (32111)  (4311)   (4321)
                                         (41111)  (32211)  (5221)
                                                  (42111)  (5311)
                                                           (32221)
                                                           (33211)
                                                           (42211)
                                                           (43111)
                                                           (331111)
                                                           (421111)
                                                           (511111)

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394* A367398
strict: A367214 A367215 A367220 A367221 A367395 A367399
subsets: A367216 A367217 A367222 A367223 A367396 A367400
ranks: A367224 A367225 A367226 A367227 A367397 A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A008284 counts partitions by length, strict A008289.
A365543 counts partitions with a subset-sum k, strict A365661.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A088809, A093971, A126796, A238628, A304792, A363225, A364534, A365541, A365924, A367402.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,10}]

A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 5, 7, 8, 11, 13, 17, 19, 25, 28, 35, 41, 49, 57, 68, 78, 92, 107, 124, 143, 166, 192, 220, 254, 291, 335, 382, 439, 499, 572, 649, 741, 840, 956, 1080, 1226, 1383, 1566, 1762, 1988, 2235, 2515, 2822, 3166, 3547
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2023

Keywords

Examples

			The strict partition (5,3,2,1) has 4 = 3 + 1 so is counted under a(11).
The a(6) = 1 through a(17) = 7 strict partitions (A..E = 10..14):
  321  421  521  621  721   821   921   A21   B21   C21    D21    E21
                      4321  5321  6321  5431  6431  6531   7531   7631
                                        7321  8321  7431   8431   8531
                                                    9321   A321   9431
                                                    54321  64321  B321
                                                                  65321
                                                                  74321

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394 A367398
strict: A367214 A367215 A367220 A367221 A367395* A367399
subsets: A367216 A367217 A367222 A367223 A367396 A367400
ranks: A367224 A367225 A367226 A367227 A367397 A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A088809/A093971 count twofold sum-full subsets.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A238628, A363225, A364272, A364534, A365661, A365925, A367410, A367411.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,30}]

A367396 Number of subsets of {1..n} whose cardinality is the sum of two distinct elements.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 17, 40, 90, 199, 435, 939, 2007, 4258, 8976, 18817, 39263, 81595, 168969, 348820, 718134, 1474863, 3022407, 6181687, 12621135, 25727686, 52369508, 106460521, 216162987, 438431215, 888359841, 1798371648, 3637518354, 7351824439, 14848255803
Offset: 0

Views

Author

Gus Wiseman, Nov 21 2023

Keywords

Examples

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is not counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                    {1,2,4}    {1,2,4}      {1,2,4}
                    {1,2,3,4}  {1,2,5}      {1,2,5}
                               {1,2,3,4}    {1,2,6}
                               {1,2,3,5}    {1,2,3,4}
                               {1,3,4,5}    {1,2,3,5}
                               {1,2,3,4,5}  {1,2,3,6}
                                            {1,3,4,5}
                                            {1,3,4,6}
                                            {1,3,5,6}
                                            {1,2,3,4,5}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394 A367398
strict: A367214 A367215 A367220 A367221 A367395 A367399
subsets: A367216 A367217 A367222 A367223 A367396* A367400
ranks: A367224 A367225 A367226 A367227 A367397 A367401
A002865 counts partitions whose length is a part, complement A229816.
A364534 counts sum-full subsets.
A088809 and A093971 count subsets containing semi-sums.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A236912, A237113, A237667, A237668, A304792, A363225, A364272, A365658, A366131, A366740.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],Length[#]]&]],{n,0,10}]
  • Python
    from itertools import combinations
def A367396(n): return sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Formula

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - 5*a(n-4) + 2*a(n-5) for n > 4.
G.f.: x^3*(x - 1)/((2*x - 1)*(x^4 - 2*x^3 + x^2 - 2*x + 1)). (End)

Extensions

a(18)-a(33) from Chai Wah Wu, Nov 21 2023
a(34) from Paul Muljadi, Nov 24 2023

A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

Original entry on oeis.org

4, 12, 18, 30, 36, 40, 42, 54, 60, 66, 78, 81, 90, 100, 102, 112, 114, 120, 126, 135, 138, 140, 150, 168, 174, 180, 186, 189, 198, 210, 220, 222, 225, 234, 246, 250, 252, 258, 260, 270, 280, 282, 297, 300, 306, 315, 318, 330, 336, 340, 342, 350, 351, 352, 354
Offset: 1

Views

Author

Gus Wiseman, Nov 21 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are the Heinz numbers of the partitions counted by A367394.

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394 A367398
strict: A367214 A367215 A367220 A367221 A367395 A367399
subsets: A367216 A367217 A367222 A367223 A367396 A367400
ranks: A367224 A367225 A367226 A367227 A367397* A367401
A325761 ranks partitions whose length is a part, counted by A002865.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A304792 counts subset-sums of partitions, strict A365925.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A229816, A237667, A237668, A238628, A363225, A364272, A365543, A365658, A365918, A366740.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MemberQ[Total/@Subsets[prix[#],{2}],PrimeOmega[#]]&]

A367398 Number of integer partitions of n whose length is not a semi-sum of the parts.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 8, 12, 16, 23, 28, 41, 52, 71, 89, 122, 151, 200, 246, 321, 398, 510, 620, 794, 968, 1212, 1474, 1837, 2219, 2748, 3302, 4055, 4882, 5942, 7094, 8623, 10275, 12376, 14721, 17661, 20920, 25011, 29516, 35120, 41419, 49053, 57609, 68092, 79780
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2023

Keywords

Comments

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Examples

			For the partition y = (4,3,1) we have semi-sums {4,5,7}, which do not include 3 (the length of y), so y is counted under a(8).
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (21)   (22)    (32)     (33)      (43)       (44)
            (111)  (31)    (41)     (42)      (52)       (53)
                   (1111)  (311)    (51)      (61)       (62)
                           (2111)   (222)     (322)      (71)
                           (11111)  (411)     (331)      (332)
                                    (21111)   (511)      (422)
                                    (111111)  (4111)     (431)
                                              (22111)    (611)
                                              (31111)    (4211)
                                              (211111)   (5111)
                                              (1111111)  (22211)
                                                         (221111)
                                                         (311111)
                                                         (2111111)
                                                         (11111111)

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394 A367398*
strict: A367214 A367215 A367220 A367221 A367395 A367399
subsets: A367216 A367217 A367222 A367223 A367396 A367400
ranks: A367224 A367225 A367226 A367227 A367397 A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367402 counts partitions with covering semi-sums, complement A367403.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A126796, A304792, A363225, A364272, A365543.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#,{2}],Length[#]]&]],{n,0,10}]

A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 13, 15, 19, 22, 27, 31, 38, 43, 51, 59, 70, 79, 94, 107, 124, 143, 165, 188, 218, 248, 283, 324, 369, 419, 476, 540, 610, 691, 778, 878, 987, 1111, 1244, 1399, 1563, 1750, 1954, 2184, 2432, 2714, 3016, 3358, 3730, 4143
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2023

Keywords

Examples

			The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
  (6)    (7)    (8)      (9)      (10)     (11)     (12)       (13)
  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)    (7,5)      (7,6)
  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)    (7,4)    (8,4)      (8,5)
         (6,1)  (7,1)    (7,2)    (8,2)    (8,3)    (9,3)      (9,4)
                (4,3,1)  (8,1)    (9,1)    (9,2)    (10,2)     (10,3)
                         (4,3,2)  (5,3,2)  (10,1)   (11,1)     (11,2)
                         (5,3,1)  (5,4,1)  (5,4,2)  (5,4,3)    (12,1)
                                  (6,3,1)  (6,3,2)  (6,4,2)    (6,4,3)
                                           (6,4,1)  (6,5,1)    (6,5,2)
                                           (7,3,1)  (7,3,2)    (7,4,2)
                                                    (7,4,1)    (7,5,1)
                                                    (8,3,1)    (8,3,2)
                                                    (5,4,2,1)  (8,4,1)
                                                               (9,3,1)
                                                               (6,4,2,1)

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394 A367398
strict: A367214 A367215 A367220 A367221 A367395 A367399*
subsets: A367216 A367217 A367222 A367223 A367396 A367400
ranks: A367224 A367225 A367226 A367227 A367397 A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A365924 counts incomplete partitions, strict A365831.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367403 counts partitions without covering semi-sums, strict A367411.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A126796, A188431, A238628, A237113, A363225, A364272, A365658, A365918, A365925, A367410.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,15}]

A367400 Number of subsets of {1..n} whose cardinality is not the sum of two distinct elements.

Original entry on oeis.org

1, 2, 4, 7, 13, 25, 47, 88, 166, 313, 589, 1109, 2089, 3934, 7408, 13951, 26273, 49477, 93175, 175468, 330442, 622289, 1171897, 2206921, 4156081, 7826746, 14739356, 27757207, 52272469, 98439697, 185381983, 349112000, 657448942, 1238110153
Offset: 0

Views

Author

Gus Wiseman, Nov 21 2023

Keywords

Examples

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is counted under a(8).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,3,4}
                         {2,3,4}

Crossrefs

The version containing n appears to be A112575.
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
-----------------------------------------------------------
partitions: A367212 A367213 A367218 A367219 A367394 A367398
strict: A367214 A367215 A367220 A367221 A367395 A367399
subsets: A367216 A367217 A367222 A367223 A367396 A367400*
ranks: A367224 A367225 A367226 A367227 A367397 A367401
A002865 counts partitions whose length is a part, complement A229816.
A364534 counts sum-full subsets.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A237113, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366131, A366740.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n,0,10}]
  • Python
    from itertools import combinations
def A367400(n): return (n*(n+1)>>1)+1+sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if not any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Formula

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) for n > 3.
G.f.: (-x^3 + x^2 + 1)/(x^4 - 2*x^3 + x^2 - 2*x + 1). (End)

Extensions

a(18)-a(33) from Chai Wah Wu, Nov 21 2023
Showing 1-7 of 7 results.

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