A237113 -id:A237113 - cp's OEIS frontend

A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

1, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 51, 60, 71, 83, 97, 113, 132, 153, 178, 205, 238, 272, 315, 360, 413, 471, 539, 613, 698, 792, 899, 1018, 1153, 1302, 1470, 1658, 1867, 2100, 2362, 2652, 2974, 3335, 3734, 4178, 4672

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367224 /\ A005117.

			The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
The a(1) = 1 through a(10) = 5 strict partitions:
  (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                           (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                             (5,2,1)  (5,3,1)  (6,3,1)
                                                      (6,2,1)  (7,2,1)
                                                               (4,3,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214*   A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A275972 counts strict knapsack partitions, non-strict A108917.
A364272 counts sum-full strict partitions, sum-free A364349.
A365925 counts subset-sums of strict partitions, non-strict A304792.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.

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

A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

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 A367218.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226*   A367227
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict partitions, counted by A000009.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A066208 ranks partitions into odd parts, counted by A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A237668 counts sum-full partitions, ranks A364532.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A088314, A106529, A116861, A237113, A238628, A299702, A364347, A365073, A367107.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]!={}&]

A364915 Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 12, 10, 16, 16, 19, 21, 29, 25, 37, 35, 44, 46, 60, 55, 75, 71, 90, 90, 114, 110, 140, 138, 167, 163, 217, 201, 248, 241, 298, 303, 359, 355, 425, 422, 520, 496, 594, 603, 715, 706, 834, 826, 968, 972, 1153, 1147, 1334, 1315, 1530

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

			The a(1) = 1 through a(10) = 8 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     52       53        72         64
                           111111  322      332       333        73
                                   1111111  2222      522        433
                                            11111111  3222       3322
                                                      111111111  22222
                                                                 1111111111
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is counted under a(12).
The partition (6,4,3,2) has 6=4+2, or 6=3+3, or 6=2+2+2, or 4=2+2, so is not counted under a(15).

For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The strict case is A364350.
The complement is A365068, strict A364839.
The positive case is A365072, strict A365006.
A000041 counts integer partitions, strict A000009.
A007865 counts binary sum-free sets w/ re-usable parts, complement A093971.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364912 counts linear combinations of partitions of k.
Cf. A085489, A108917, A151897, A237113, A323092, A364272, A364533, A364910, A364911, A364913.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], Function[ptn,!Or@@Table[combs[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]], {n,0,15}]

from sympy.utilities.iterables import partitions
def A364915(n):
    if n <= 1: return 1
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
    for p in partitions(n,k=n-1):
        s = set(p)
        if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
            c += 1
    return c # Chai Wah Wu, Sep 23 2023

a(n) = A000041(n) - A365068(n).

a(37)-a(59) from Chai Wah Wu, Sep 25 2023

A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 21, 28, 35, 46, 57, 70, 87, 110, 130, 165, 198, 238, 285, 349, 410, 498, 583, 702, 819, 983, 1136, 1353, 1570, 1852, 2137, 2520, 2898, 3390, 3891, 4540, 5191, 6028, 6889, 7951, 9082, 10450, 11884, 13650, 15508, 17728, 20113

Offset: 0

Gus Wiseman, Jul 19 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

For length instead of differences we have A229816, strict A240861.
For all differences of pairs parts we have A364345.
For subsets of {1..n} instead of partitions we have A364463.
The strict case is A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first-differences.
Cf. A002865, A025065, A108917, A236912, A237113, A237667, A320347, A326083, A363225, A364347.

Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]

from collections import Counter
from sympy.utilities.iterables import partitions
def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), 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

A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Also Heinz numbers of a type of sum-free partitions not allowing re-used parts, counted by A236912.

			The prime indices of 198 are {1,2,2,5}, which is sum-free even though it is not knapsack (A299702, A299729), so 198 is in the sequence.

Subsets of this type are counted by A085489, with re-usable parts A007865.
Subsets not of this type are counted by A093971, w/ re-usable parts A088809.
Partitions of this type are counted by A236912.
Allowing parts to be re-used gives A364347, counted by A364345.
The complement allowing parts to be re-used is A364348, counted by A363225.
The non-binary version allowing re-used parts is counted by A364350.
The complement is A364462, counted by A237113.
The non-binary version is A364531, counted by A237667, complement A364532.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A288728, A320340, A325862, A325864, A326083, A363226, A364346.

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

A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

Original entry on oeis.org

12, 24, 30, 36, 48, 60, 63, 70, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 154, 156, 165, 168, 180, 189, 192, 204, 210, 216, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 324, 325, 330, 336, 348, 350, 360, 372, 378, 384, 390

Offset: 1

Gus Wiseman, Jul 29 2023

nonn

Also Heinz numbers of a type of sum-full partitions not allowing re-used parts, counted by A237113.
No partitions of this type are knapsack (A299702, A299729).
All multiples of terms are terms. - Robert Israel, Aug 30 2023

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   72: {1,1,1,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}

Robert Israel, Table of n, a(n) for n = 1..10000

Subsets not of this type are counted by A085489, w/ re-usable parts A007865.
Subsets of this type are counted by A088809, with re-usable parts A093971.
Partitions not of this type are counted by A236912.
Partitions of this type are counted by A237113.
Subset of A299729.
The complement with re-usable parts is A364347, counted by A364345.
With re-usable parts we have A364348, counted by A363225 (strict A363226).
The complement is A364461.
The non-binary complement is A364531, counted by A237667.
The non-binary version is A364532, see also A364350.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A320340, A325862, A326083.

filter:= proc(n) local F, i,j,m;
  F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
  for i from 1 to nops(F)-1 do for j from 1 to i-1 do
    if member(F[i]+F[j],F) then return true fi
  od od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 30 2023

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

A364911 Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 5, 2, 5, 3, 3, 1, 6, 3, 8, 4, 4, 4, 1, 7, 3, 11, 6, 6, 6, 5, 1, 8, 4, 14, 9, 8, 10, 7, 6, 1, 9, 4, 19, 11, 11, 14, 11, 9, 8, 1, 10, 5, 23, 14, 15, 21, 15, 14, 11, 10, 1, 11, 5, 28, 17, 19, 28, 22, 20, 17, 15, 12

Offset: 0

Gus Wiseman, Aug 27 2023

Also the number of ways to write any number up to n as a positive linear combination of a strict integer partition of k.

			Triangle begins:
  1
  1  1
  1  2  1
  1  3  1  2
  1  4  2  3  2
  1  5  2  5  3  3
  1  6  3  8  4  4  4
  1  7  3 11  6  6  6  5
  1  8  4 14  9  8 10  7  6
  1  9  4 19 11 11 14 11  9  8
  1 10  5 23 14 15 21 15 14 11 10
  1 11  5 28 17 19 28 22 20 17 15 12
  1 12  6 34 21 22 40 28 28 24 24 17 15
  1 13  6 40 25 27 50 38 37 34 35 27 22 18
  1 14  7 46 29 32 65 49 50 43 51 38 35 26 22
  1 15  7 54 33 38 79 62 63 59 68 55 50 41 32 27
Row n = 5 counts the following partitions:
    .    1           2     3         4       5
         1+1         2+2   1+2       1+3     1+4
         1+1+1             1+1+2     1+1+3   2+3
         1+1+1+1           1+1+1+2
         1+1+1+1+1         1+2+2
Row n = 5 counts the following positive linear combinations:
  .  1*1  1*2  1*3      1*4      1*5
     2*1  2*2  1*2+1*1  1*3+1*1  1*3+1*2
     3*1       1*2+2*1  1*3+2*1  1*4+1*1
     4*1       1*2+3*1
     5*1       2*2+1*1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column n = k is A000009.
Column k = 0 is A000012.
Column k = 1 is A000027.
Row sums are A000070.
Column k = 2 is A008619.
Columns are partial sums of columns of A116861.
Column k = 3 appears to be the partial sums of A137719.
Diagonal n = 2k is A364910.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A114638 counts partitions where (length) = (sum of distinct parts).
A116608 counts partitions by number of distinct parts.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002865, A066328, A179009, A236912, A237113, A237667, A364912, A364913, A364915, A364916, A365002, A365004.

Table[Length[Select[Array[IntegerPartitions,n+1,0,Join],Total[Union[#]]==k&]],{n,0,9},{k,0,n}]

T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

G.f.: A(x,y) = (1/(1 - x)) * Product_{k>=1} (1 - y^k + y^k/(1 - x^k)). - Andrew Howroyd, Jan 11 2024

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

A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 21, 32, 49, 70, 99, 135, 185, 245, 323, 418, 541, 688, 873, 1094, 1368, 1693, 2092, 2564, 3138, 3810, 4620, 5565, 6696, 8012, 9569, 11381, 13518, 15980, 18872, 22194, 26075, 30535, 35711, 41627, 48473, 56290, 65283, 75533, 87298, 100631, 115911, 133219

Offset: 0

Gus Wiseman, Aug 25 2023

nonn

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} such that some element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,4,9} has 9 = 3*3 + 0*4, but this is not strictly positive, so S is not counted under a(9).
The subset S = {3,4,10} has 10 = 2*3 + 1*4, so S is counted under a(10).
The a(0) = 0 through a(5) = 12 subsets:
  .  .  {1,2}  {1,2}    {1,2}    {1,2}
               {1,3}    {1,3}    {1,3}
               {1,2,3}  {1,4}    {1,4}
                        {2,4}    {1,5}
                        {1,2,3}  {2,4}
                        {1,2,4}  {1,2,3}
                        {1,3,4}  {1,2,4}
                                 {1,2,5}
                                 {1,3,4}
                                 {1,3,5}
                                 {1,4,5}
                                 {2,3,5}

Bert Dobbelaere, Table of n, a(n) for n = 0..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary complement is A007865, first differences A288728.
The binary version is A093971, first differences A365070.
The nonnegative complement is A326083, first differences A124506.
The nonnegative version is A364914, first differences A365046.
First differences are A365042.
The complement is counted by A365044, first differences A365045.
Without re-usable parts we have A364534, first differences A365069.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088809 and A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Rest[Subsets[Range[n]]],combp[Last[#],Union[Most[#]]]!={}&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365043(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365044(n).

a(15)-a(35) from Chai Wah Wu, Nov 20 2023

More terms from Bert Dobbelaere, Apr 28 2025

A365312 Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 2, 6, 4, 8, 7, 16, 6, 24, 17, 24, 20, 46, 22, 62, 31, 63, 57, 106, 35, 122, 90, 137, 88, 212, 74, 262, 134, 267, 206, 345, 121, 476, 294, 484, 232, 698, 242, 837, 389, 763, 571, 1185, 318, 1327, 634, 1392, 727, 1927, 640, 2056, 827, 2233, 1328

Offset: 0

Gus Wiseman, Sep 05 2023

nonn

			The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
The a(0) = 0 through a(11) = 16 strict partitions:
  .  .  .  (2)  (3)  (2)  (4)  (2)    (3)  (2)    (3)    (2)
                     (3)  (5)  (3)    (5)  (4)    (4)    (3)
                     (4)       (4)    (6)  (5)    (6)    (4)
                               (5)    (7)  (6)    (7)    (5)
                               (6)         (7)    (8)    (6)
                               (4,2)       (8)    (9)    (7)
                                           (4,2)  (6,3)  (8)
                                           (6,2)         (9)
                                                         (10)
                                                         (4,2)
                                                         (5,4)
                                                         (6,2)
                                                         (6,3)
                                                         (6,4)
                                                         (7,3)
                                                         (8,2)

The complement for positive coefficients is counted by A088314.
For positive coefficients we have A088528.
The complement is counted by A365311.
For non-strict partitions we have A365378, complement A365379.
The version for subsets is A365380, complement A365073.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&],combs[n,#]=={}&]],{n,0,10}]

from math import isqrt
from sympy.utilities.iterables import partitions
def A365312(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(26)-a(58) from Chai Wah Wu, Sep 13 2023

cp's OEIS Frontend

A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

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A364915 Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.

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A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

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A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

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A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

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Maple

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A364911 Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.

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

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