A323092 -id:A323092 - cp's OEIS frontend

A308546 Number of double-closed subsets of {1..n}.

1, 2, 3, 6, 8, 16, 24, 48, 60, 120, 180, 360, 480, 960, 1440, 2880, 3456, 6912, 10368, 20736, 27648, 55296, 82944, 165888, 207360, 414720, 622080, 1244160, 1658880, 3317760, 4976640, 9953280, 11612160, 23224320, 34836480, 69672960, 92897280

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

These are subsets containing twice any element whose double is <= n.
Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:
  {}  {1}  {1,2}  {1,2,3}  {1,2,3,4}  {1,2,3,4,5}  {1,2,3,4,5,6}
      {3}  {1,3}  {1,2,4}  {1,2,3,5}  {1,2,3,4,6}
      {5}  {1,5}  {1,2,5}  {1,2,3,6}  {1,2,3,5,6}
           {3,5}  {1,3,5}  {1,2,4,5}
           {3,6}  {1,3,6}  {1,3,5,6}
                  {3,5,6}

			The a(6) = 24 subsets:
  {}  {4}  {2,4}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {5}  {3,6}  {2,4,5}  {1,2,4,6}  {1,2,4,5,6}
      {6}  {4,5}  {2,4,6}  {2,3,4,6}  {2,3,4,5,6}
           {4,6}  {3,4,6}  {2,4,5,6}
           {5,6}  {3,5,6}  {3,4,5,6}
                  {4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..500

Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[2*#,#<=n&]]&]],{n,0,10}]

From Charlie Neder, Jun 10 2019: (Start)
a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).
a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)

a(21)-a(36) from Charlie Neder, Jun 10 2019

A101417 Number of partitions of n into parts without powers of 2.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 3, 6, 5, 6, 10, 9, 12, 17, 17, 22, 28, 30, 37, 48, 52, 62, 78, 86, 103, 127, 141, 166, 201, 227, 266, 317, 358, 417, 492, 560, 647, 757, 860, 991, 1153, 1309, 1503, 1738, 1971, 2257, 2594, 2941, 3356, 3843, 4351, 4948, 5644, 6382, 7240

Offset: 0

Reinhard Zumkeller, Jan 16 2005

nonn

			a(12) = #{3+3+3+3, 6+3+3, 6+6, 7+5, 9+3, 12} = 6.
From _Gus Wiseman_, Jan 07 2019: (Start)
The a(3) = 1 through a(14) = 5 integer partitions (A = 10, ..., E = 14):
  (3)  (5)  (6)   (7)  (53)  (9)    (A)   (B)    (C)     (D)    (E)
            (33)             (63)   (55)  (65)   (66)    (76)   (77)
                             (333)  (73)  (533)  (75)    (A3)   (95)
                                                 (93)    (553)  (B3)
                                                 (633)   (733)  (653)
                                                 (3333)         (5333)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A018819, A000123.

Cf. A087897, A102430, A276431, A321346, A323053, A323092, A323093.

g:= product(1-x^(2^j),j=0..15)/product(1-x^i,i=1..75): gser:= series(g, x=0,62): seq(coeff(gser,x,n),n=0..59); # Emeric Deutsch, Mar 29 2006

Table[Length[Select[IntegerPartitions[n],And@@Not/@IntegerQ/@Log[2,#]&]],{n,20}] (* Gus Wiseman, Jan 07 2019 *)

G.f.: Product_{j>=1} (1-x^(2^j)) / Product_{i>=2} (1-x^i). - Emeric Deutsch, Mar 29 2006

A364910 Number of integer partitions of 2n whose distinct parts sum to n.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 12, 11, 19, 23, 54, 55, 103, 115, 178, 289, 389, 507, 757, 970, 1343, 2033, 2579, 3481, 4840, 6312, 8317, 10998, 15459, 19334, 26368, 33480, 44709, 56838, 74878, 93369, 128109, 157024, 206471, 258357, 338085, 417530, 544263, 669388, 859570, 1082758, 1367068

Offset: 0

Gus Wiseman, Aug 16 2023

nonn

Also the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of n.

			The a(0) = 1 through a(7) = 11 partitions:
  ()  (11)  (22)  (33)     (44)      (55)       (66)         (77)
                  (2211)   (3311)    (3322)     (4422)       (4433)
                  (21111)  (311111)  (4411)     (5511)       (5522)
                                     (4111111)  (33321)      (6611)
                                                (42222)      (442211)
                                                (322221)     (4222211)
                                                (332211)     (4421111)
                                                (3222111)    (42221111)
                                                (3321111)    (422111111)
                                                (32211111)   (611111111)
                                                (51111111)   (4211111111)
                                                (321111111)
The a(0) = 1 through a(7) = 11 linear combinations:
  0  1*1  1*2  1*3      1*4      1*5      1*6          1*7
               0*2+3*1  0*3+4*1  0*4+5*1  0*4+3*2      0*6+7*1
               1*2+1*1  1*3+1*1  1*3+1*2  0*5+6*1      1*4+1*3
                                 1*4+1*1  1*4+1*2      1*5+1*2
                                          1*5+1*1      1*6+1*1
                                          0*3+0*2+6*1  0*4+0*2+7*1
                                          0*3+1*2+4*1  0*4+1*2+5*1
                                          0*3+2*2+2*1  0*4+2*2+3*1
                                          0*3+3*2+0*1  0*4+3*2+1*1
                                          1*3+0*2+3*1  1*4+0*2+3*1
                                          1*3+1*2+1*1  1*4+1*2+1*1
                                          2*3+0*2+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500 (first 91 terms from David A. Corneth)

The case with no zero coefficients is A000009.
Central diagonal of A116861.
A version based on Heinz numbers is A364906.
Using all partitions (not just strict) we get A364907.
The version for compositions is A364908, strict A364909.
Main diagonal of A364916.
Using strict partitions of any number from 1 to n gives A365002.
These partitions have ranks A365003.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A237113, A364350, A364839, A364911, A364912, A364914.

Table[Length[Select[IntegerPartitions[2n],Total[Union[#]]==n&]],{n,0,15}]

a(n) = {my(res = 0); forpart(p = 2*n,s = Set(p); if(vecsum(s) == n, res++)); res} \\ David A. Corneth, Aug 20 2023

from sympy.utilities.iterables import partitions
def A364910(n): return sum(1 for d in partitions(n<<1,k=n) if sum(set(d))==n) # Chai Wah Wu, Sep 13 2023

a(n) = A116861(2n,n).

a(n) = A364916(n,n).

More terms from David A. Corneth, Aug 20 2023

A364348 Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Or numbers with a prime index equal to the sum of two others, allowing re-used parts.

Also Heinz numbers of a type of sum-free partitions counted by A363225.

			We have 6 because prime(1) and prime(1) are both divisors of 6, and prime(1+1) is also.
The terms together with their prime indices begin:
   6: {1,2}
  12: {1,1,2}
  18: {1,2,2}
  21: {2,4}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  70: {1,3,4}
  72: {1,1,1,2,2}

Subsets of this type are counted by A093971, complement A007865.
Partitions of this type are counted by A363225, strict A363226.
The complement is A364347, counted by A364345.
The complement without re-using parts is A364461, counted by A236912.
Without re-using parts we have A364462, counted by A237113.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
A323092 counts double-free partitions, ranks A320340.
Cf. A237667, A275972, A288728, A325862, A326083, A364346.

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

A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 15, 21, 22, 28, 32, 38, 40, 51, 55, 65, 74, 83, 94, 111, 119, 136, 160, 174, 196, 222, 252, 273, 315, 341, 391, 425, 477, 518, 602, 636, 719, 782, 886, 944, 1073, 1140, 1302, 1380, 1553, 1651, 1888, 1995, 2224, 2370

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4    5    6    7     8     9     A     B     C
          21   31   32   42   43    53    54    64    65    75
                    41   51   52    62    63    73    74    84
                              61    71    72    82    83    93
                              421   521   81    91    92    A2
                                          432   631   A1    B1
                                          531   721   542   543
                                          621         632   732
                                                      641   741
                                                      731   831
                                                      821   921

For subsets of {1..n} we have A085489, complement A088809.
The non-strict version is A236912, complement A237113, ranked by A364461.
Allowing re-used parts gives A364346.
The non-binary version is A364349, non-strict A237667 (complement A237668).
The linear combination-free version is A364350.
The complement in strict partitions is A364670, w/ re-used parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972.
A151897 counts sum-free subsets, complement A364534.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A240861, A325862, A326083, A364272, A364345, A364347.

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

A350844 Number of strict integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867

Offset: 0

Gus Wiseman, Jan 21 2022

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4   5    6     7    8     9     A      B     C
          21       32   51    43   62    54    73     65    84
                   41   321   52   71    63    82     74    93
                              61   521   72    91     83    A2
                                         81    541    92    B1
                                         432   721    A1    543
                                         621   4321   632   651
                                                      821   732
                                                            741
                                                            921
                                                            6321

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for no difference 0 is A000009.
The version for no difference > -2 is A001227, non-strict A034296.
The version for no difference -1 is A003114 (A325160).
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The opposite version is A072670.
The multiplicative version is A350840, non-strict A350837 (A350838).
The non-strict version is A350842.
A000041 counts integer partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length (A026424).
A116931 counts partitions with no difference -1 (A319630).
A323092 counts double-free integer partitions (A320340) strict A120641.
A325534 counts separable partitions (A335433).
A325535 counts inseparable partitions (A335448).
Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],0|-2]&]],{n,0,30}]

A364463 Number of subsets of {1..n} with elements disjoint from first differences of elements.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 30, 54, 92, 167, 290, 525, 935, 1704, 3082, 5664, 10386, 19249, 35701, 66702, 124855, 234969, 443174, 839254, 1592925, 3032757, 5786153, 11066413, 21204855, 40712426, 78294085, 150815154, 290922900, 561968268, 1086879052, 2104570243

Offset: 0

Gus Wiseman, Jul 27 2023

nonn

In other words, no element is the difference of two consecutive elements.
From David A. Corneth, Aug 02 2023: (Start)
As subsets counted in a(n) are also counted in a(n+1) and {n+1} is a subset counted in a(n+1) but not a(n), a(n + 1) > a(n) for n >= 1.
As every subset counted in a(n + 1) that contains n+1 can be found from some subset counted in a(n) by appending n+1 and every subset counted in a(n) not containing n + 1 is counted in a(n + 1), a(n+1) <= 2*a(n). (End)

			The a(0) = 1 through a(5) = 18 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                {2,3}  {1,3}    {5}
                       {1,4}    {1,3}
                       {2,3}    {1,4}
                       {3,4}    {1,5}
                       {2,3,4}  {2,3}
                                {2,5}
                                {3,4}
                                {3,5}
                                {4,5}
                                {1,3,5}
                                {2,3,4}
                                {3,4,5}
                                {2,3,4,5}

For all differences of pairs of elements we have A007865.
For partitions instead of subsets we have A363260, strict A364464.
The complement is counted by A364466.
A000041 counts integer partitions, strict A000009.
A364465 counts subsets with distinct first differences, partitions A325325.
Cf. A011782, A025065, A229816, A236912, A237113, A237667, A240861, A320347, A323092, A326083, A364347.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]=={}&]],{n,0,10}]

from itertools import combinations
def A364463(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

a(n) < a(n + 1) <= 2 * a(n). - David A. Corneth, Aug 02 2023

a(21)-a(29) from David A. Corneth, Aug 02 2023
a(30)-a(32) from Chai Wah Wu, Sep 26 2023
a(33)-a(35) from Chai Wah Wu, Sep 27 2023

A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 9, 12, 13, 18, 23, 29, 37, 49, 55, 71, 84, 104, 126, 153, 185, 221, 261, 317, 375, 446, 523, 623, 721, 854, 994, 1168, 1357, 1579, 1833, 2126, 2455, 2843, 3270, 3766, 4320, 4980, 5687, 6521, 7444, 8498, 9684, 11039, 12540, 14262

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(1) = 1 through a(8) = 12 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (311)    (51)      (52)       (53)
                    (1111)  (11111)  (222)     (61)       (62)
                                     (3111)    (322)      (71)
                                     (111111)  (331)      (332)
                                               (511)      (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (311111)
                                                          (11111111)

Cf. A018819, A101417, A120641, A276431, A305148, A323092, A323094.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[#,IntegerQ[Log[2,#1/#2]]&]&]],{n,30}]

A364464 Number of strict integer partitions of n where no part is the difference of two consecutive parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 4, 6, 5, 8, 9, 12, 13, 16, 16, 21, 23, 29, 34, 38, 41, 49, 57, 64, 73, 86, 95, 110, 120, 135, 160, 171, 197, 219, 247, 277, 312, 342, 386, 431, 476, 527, 598, 640, 727, 796, 893, 966, 1097, 1178, 1327, 1435, 1602, 1740, 1945, 2084, 2337

Offset: 0

Gus Wiseman, Jul 30 2023

nonn

In other words, the parts are disjoint from the first differences.

			The strict partition y = (9,5,3,1) has differences (4,2,2), and these are disjoint from the parts, so y is counted under a(18).
The a(1) = 1 through a(9) = 6 strict partitions:
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)
                 (3,1)  (3,2)  (5,1)  (4,3)  (5,3)  (5,4)
                        (4,1)         (5,2)  (6,2)  (7,2)
                                      (6,1)  (7,1)  (8,1)
                                                    (4,3,2)
                                                    (5,3,1)

For length instead of differences we have A240861, non-strict A229816.
For all differences of pairs of elements we have A364346, for subsets A007865.
For subsets instead of strict partitions we have A364463, complement A364466.
The non-strict version is A363260.
The complement is counted by A364536, non-strict A364467.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A120641 counts strict double-free partitions, non-strict A323092.
A320347 counts strict partitions w/ distinct differences, non-strict A325325.
Cf. A002865, A025065, A236912, A237667, A275972, A363226, A364345, A364465.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]

from collections import Counter
from sympy.utilities.iterables import partitions
def A364464(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(),default=1)==1,partitions(n,size=True))) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

A364670 Number of strict integer partitions of n with a part equal to the sum of two distinct others. A variation of sum-full strict partitions.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 7, 6, 10, 10, 14, 16, 24, 25, 34, 39, 48, 59, 71, 81, 103, 120, 136, 166, 194, 226, 260, 312, 353, 419, 473, 557, 636, 742, 824, 974, 1097, 1266, 1418, 1646, 1837, 2124, 2356, 2717, 3029, 3469, 3830, 4383, 4884, 5547

Offset: 0

Gus Wiseman, Aug 03 2023

nonn

			The a(6) = 1 through a(16) = 10 strict partitions (A = 10):
  321  .  431  .  532   5321  642   5431  743   6432   853
                  541         651   6421  752   6531   862
                  4321        5421  7321  761   7431   871
                              6321        5432  7521   6532
                                          6431  9321   6541
                                          6521  54321  7432
                                          8321         7621
                                                       8431
                                                       A321
                                                       64321

For subsets of {1..n} we have A088809, complement A085489.
The non-strict version is A237113, complement A236912.
The non-binary complement is A237667, ranks A364532.
Allowing re-used parts gives A363226, non-strict A363225.
The non-binary version is A364272, non-strict A237668.
The complement is A364533, non-binary A364349.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A093971, A111133, A151897, A240861, A325862, A364346, A364350, A364534.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2}]]!={}&]],{n,0,30}]

cp's OEIS Frontend

A308546 Number of double-closed subsets of {1..n}.

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A101417 Number of partitions of n into parts without powers of 2.

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A364910 Number of integer partitions of 2n whose distinct parts sum to n.

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A364348 Numbers with two possibly equal divisors prime(a) and prime(b) such that prime(a+b) is also a divisor.

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A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

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A350844 Number of strict integer partitions of n with no difference -2.

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A364463 Number of subsets of {1..n} with elements disjoint from first differences of elements.

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A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

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