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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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

A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
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 A367398.

Examples

			60 has semiprime divisor 10 with prime indices {1,3} summing to 4 = bigomega(60), so 60 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}

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 w/ length, complement A229816, ranks A325761.
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.
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, A237667, A238628, A304792, 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], FreeQ[Total/@Subsets[prix[#],{2}], PrimeOmega[#]]&]

A367097 Least positive integer whose multiset of prime indices has exactly n distinct semi-sums.

Original entry on oeis.org

1, 4, 12, 30, 60, 210, 330, 660, 2730, 3570, 6270, 12540, 53130, 79170, 110670, 221340, 514140, 1799490, 2284590, 4196010, 6750870, 13501740, 37532220, 97350330, 131362770, 189620970, 379241940, 735844830, 1471689660
Offset: 0

Views

Author

Gus Wiseman, Nov 09 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.
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.
From David A. Corneth, Nov 15 2023: (Start)
Terms are cubefree.
bigomega(a(n)) = A001222(a(n)) >= A002024(n) + 1 = floor(sqrt(2n) + 1/2) + 1 for n > 0. (End)

Examples

			The prime indices of 60 are {1,1,2,3}, with four semi-sums {2,3,4,5}, and 60 is the first number whose prime indices have four semi-sums, so a(4) = 60.
The terms together with their prime indices begin:
       1: {}
       4: {1,1}
      12: {1,1,2}
      30: {1,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
     330: {1,2,3,5}
     660: {1,1,2,3,5}
    2730: {1,2,3,4,6}
    3570: {1,2,3,4,7}
    6270: {1,2,3,5,8}
   12540: {1,1,2,3,5,8}
   53130: {1,2,3,4,5,9}
   79170: {1,2,3,4,6,10}
  110670: {1,2,3,4,7,11}
  221340: {1,1,2,3,4,7,11}
  514140: {1,1,2,3,5,8,13}

Crossrefs

The non-binary version is A259941, firsts of A299701.
These are the positions of first appearances in A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
A366738 counts semi-sums of partitions, strict A366741.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A002024, A004709, A304793, A365920, A366740, A366753, A367093, A367095.

Programs

  • Mathematica
    nn=1000;
w=Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
v=Table[Position[w,k][[1,1]],{k,0,spnm[w]}]
  • Python
    from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A367097(n): return next(k for k in count(1) if len({sum(s) for s in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)}) == n) # Chai Wah Wu, Nov 13 2023

Formula

2 | a(n) for n > 0. - David A. Corneth, Nov 13 2023

Extensions

a(17)-a(22) from Chai Wah Wu, Nov 13 2023
a(23)-a(28) from David A. Corneth, Nov 13 2023

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15
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

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Crossrefs

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

Programs

  • Mathematica
    DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]
Previous Showing 11-17 of 17 results.

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