A342191 -id:A342191 - cp's OEIS frontend

A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 7, 8, 10, 13, 17, 19, 22, 25, 30, 35, 43, 52, 60, 70, 81, 93, 106, 122, 142, 166, 190, 216, 249, 287, 325, 371, 420, 479, 543, 617, 695, 784, 888, 1000, 1126, 1266, 1420, 1594, 1792, 2008, 2247, 2514, 2809, 3135, 3496, 3891, 4332

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
  1   2   3   4    5    6    7    8     9     A     B     C     D
              31   32   51   43   53    54    64    65    75    76
                   41        52   62    72    73    74    93    85
                             61   71    81    82    83    A2    94
                                  431   432   91    92    B1    A3
                                        531   532   A1    543   B2
                                              541   641   651   C1
                                                    731   732   643
                                                          741   652
                                                          831   751
                                                                832
                                                                931
                                                                5431

The version for subsets of prescribed maximum is A045691.
The double-free case is A120641.
The non-strict case is A350837, ranked by A350838.
An additive version (differences) is A350844, non-strict A350842.
The non-strict complement is counted by A350846, ranked by A350845.
Versions for prescribed quotients:
  = 2:  A154402, sets A001511.
  != 2: A350840 (this sequence), sets A045691.
  >= 2: A000929, sets A018819.
  <= 2: A342095, non-strict A342094.
  < 2:  A342097, non-strict A342096, sets A045690.
  > 2:  A342098, sets A040039.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]

A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

Original entry on oeis.org

0, 1, 1, 3, 5, 11, 19, 41, 77, 159, 307, 625, 1231, 2481, 4921, 9883, 19689, 39455, 78751, 157661, 315015, 630337, 1260049, 2520723, 5040215, 10081661, 20160841, 40324163, 80643405, 161291731, 322573579, 645157041, 1290294393, 2580608475, 5161177495

Offset: 0

Torsten Sillke (torsten.sillke(AT)lhsystems.com)

nonn

From Gus Wiseman, Jan 22 2022: (Start)
Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:
  {1}  {2}  {3}    {4}      {5}        {6}
            {1,3}  {1,4}    {1,5}      {1,6}
            {2,3}  {3,4}    {2,5}      {2,6}
                   {1,3,4}  {3,5}      {4,6}
                   {2,3,4}  {4,5}      {5,6}
                            {1,3,5}    {1,4,6}
                            {1,4,5}    {1,5,6}
                            {2,3,5}    {2,5,6}
                            {3,4,5}    {3,4,6}
                            {1,3,4,5}  {3,5,6}
                            {2,3,4,5}  {4,5,6}
                                       {1,3,4,6}
                                       {1,3,5,6}
                                       {1,4,5,6}
                                       {2,3,4,6}
                                       {2,3,5,6}
                                       {3,4,5,6}
                                       {1,3,4,5,6}
                                       {2,3,4,5,6}
(End)

If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
- The version with quotients <= 1/2 is A018819, partitions A000929.
- The version with quotients < 1/2 is A040039, partitions A342098.
- The version with quotients >= 1/2 is A045690(n+1), partitions A342094.
- The version with quotients > 1/2 is A045690, partitions A342096.
- Partitions of this type are counted by A350837, ranked by A350838.
- Strict partitions of this type are counted by A350840.
- For differences instead of quotients we have A350842, strict A350844.
- Partitions not of this type are counted by A350846, ranked by A350845.
A000740 = relatively prime subsets of {1..n} containing n.
A002843 = compositions with all adjacent quotients >= 1/2.
A050291 = double-free subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
A323092 = double-free integer partitions, ranked by A320340, strict A120641.
A326115 = maximal double-free subsets of {1..n}.
Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[#[[i-1]]/#[[i]]!=1/2,{i,2,Length[#]}]&]],{n,0,15}] (* Gus Wiseman, Jan 22 2022 *)

a(2*n-1) = 2*a(2*n-2) - a(n) for n >= 2; a(2*n) = 2*a(2*n-1) + a(n) for n >= 2.

More terms from Sean A. Irvine, Mar 18 2021

A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 36, 37, 41, 42, 43, 47, 48, 49, 53, 54, 59, 61, 63, 64, 65, 67, 71, 72, 73, 79, 81, 83, 84, 89, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 126, 127, 128, 131, 133, 137

Offset: 1

Gus Wiseman, Mar 11 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            19: {8}             48: {1,1,1,1,2}
      2: {1}           21: {2,4}           49: {4,4}
      3: {2}           23: {9}             53: {16}
      4: {1,1}         24: {1,1,1,2}       54: {1,2,2,2}
      5: {3}           25: {3,3}           59: {17}
      6: {1,2}         27: {2,2,2}         61: {18}
      7: {4}           29: {10}            63: {2,2,4}
      8: {1,1,1}       31: {11}            64: {1,1,1,1,1,1}
      9: {2,2}         32: {1,1,1,1,1}     65: {3,6}
     11: {5}           36: {1,1,2,2}       67: {19}
     12: {1,1,2}       37: {12}            71: {20}
     13: {6}           41: {13}            72: {1,1,1,2,2}
     16: {1,1,1,1}     42: {1,2,4}         73: {21}
     17: {7}           43: {14}            79: {22}
     18: {1,2,2}       47: {15}            81: {2,2,2,2}

The first condition alone gives A000961 (perfect powers).
The second condition alone is counted by A154402.
These partitions are counted by A342337.
A018819 counts partitions into powers of 2.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A045690 counts sets with maximum n in with adjacent elements y < 2x.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
A342342 counts strict compositions with adjacent parts x <= 2y and y <= 2x.
Cf. A003114, A003242, A034296, A040039, A167606. A342083, A342084, A342087, A342191, A342336, A342339, A342340.

Select[Range[100],With[{y=PrimePi/@First/@FactorInteger[#]},And@@Table[y[[i]]==y[[i-1]]||y[[i]]==2*y[[i-1]],{i,2,Length[y]}]]&]

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset: 1

Gus Wiseman, Jan 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-up partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}

Robert Price, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A304678.
For differences instead of quotients we have A325360 (count: A240026).
These partitions are counted by A342523 (strict: A342516, ordered: A342492).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A000005, A002843, A056239, A067824, A112798, A124010, A130091, A238710, A253249, A325351, A325352, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],LessEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-down partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A242031.
For differences instead of quotients we have A325361 (count: A320466).
These partitions are counted by A342513 (strict: A342519, ordered: A069916).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],GreaterEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

cp's OEIS Frontend

A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

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A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

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A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

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A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

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A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

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A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

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