A276024 -id:A276024 - cp's OEIS frontend

A301988 Nonprime Heinz numbers of integer partitions whose product is equal to their sum.

9, 30, 84, 108, 200, 264, 624, 1120, 1440, 1632, 3648, 7040, 8832, 12544, 16128, 20736, 22272, 33280, 47616, 76800, 113664, 157696, 174080, 202752, 251904, 528384, 778240, 1155072, 1490944, 1916928, 2605056, 3440640, 3768320, 3964928, 4423680, 5799936

Offset: 1

Gus Wiseman, Mar 30 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of reversed integer partitions begins: (22), (123), (1124), (11222), (11133), (11125), (111126), (1111134), (11111223), (1111127), (11111128), (111111135), (111111129), (1111111144), (11111111224), (111111112222).

Cf. A001055, A002865, A003963, A056239, A276024, A284640, A296150, A299701, A301957, A301987.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200000],!PrimeQ[#]&&Total[primeMS[#]]===Times@@primeMS[#]&]

A305611 Number of distinct positive subset-sums of the multiset of prime factors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 9, 1, 3, 5, 6, 3, 7, 1, 5, 3, 6, 1, 10, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 10, 3, 3

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

An integer n is a positive subset-sum of a multiset y if there exists a nonempty submultiset of y with sum n.

One less than the number of distinct values obtained when A001414 is applied to all divisors of n. - Antti Karttunen, Jun 13 2018

			The a(12) = 5 positive subset-sums of {2, 2, 3} are 2, 3, 4, 5, and 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A027746, A056239, A122768, A276024, A284640, A299701, A299702, A301855, A301935, A301957, A304792, A304793, A305613.

Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[p,{k}]]]]]],{n,100}]

up_to = 65537;
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
v001414 = vector(up_to,n,A001414(n));
A305611(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v001414[d]), mapput(m,s,s); k++)); (k-1); }; \\ Antti Karttunen, Jun 13 2018

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A305611(n):
    fs = factorint(n)
    return len(set(sum(d) for i in range(1,sum(fs.values())+1) for d in multiset_combinations(fs,i))) # Chai Wah Wu, Aug 23 2021

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 1, 1, 0, 1, 1, 2, 2, 4, 3, 2, 1, 1, 0, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 0, 1, 1, 3, 3, 4, 6, 4, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 6, 4

Offset: 0

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  1  2  1  1
  0  1  1  1  2  1  1
  0  1  1  2  3  2  1  1
  0  1  1  2  1  3  2  1  1
  0  1  1  2  2  4  3  2  1  1
  0  1  1  2  3  1  4  3  2  1  1
  0  1  1  3  3  4  6  4  3  2  1  1
  0  1  1  1  1  3  1  6  4  3  2  1  1
  0  1  1  3  3  5  4  7  6  4  3  2  1  1
  0  1  1  2  3  5  4  1  7  6  4  3  2  1  1
  0  1  1  2  3  4  6  6 11  7  6  4  3  2  1  1
Row n = 9 counts the following partitions:
  (111111111)  (22221)  (333)   (432)  (54)     (63)    (72)   (81)  (9)
                        (3222)  (441)  (522)    (621)   (711)
                                       (531)    (6111)
                                       (51111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.

Row sums are A108917.
Column k = 3 is A326034.
Cf. A002033, A196723, A275972, A276024.
Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 0, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 2, 5, 0, 29, 2, 9, 8, 20, 2

Offset: 1

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

The Heinz numbers of these partitions are given by A326018.

			The initial terms count the following partitions:
  15: (5,4,3,3)
  21: (7,6,5,3)
  21: (7,5,3,3,3)
  24: (8,7,6,3)
  25: (7,5,5,4,4)
  27: (9,8,7,3)
  27: (9,7,6,5)
  27: (8,7,3,3,3,3)
  31: (10,8,6,6,1)
  33: (11,9,7,3,3)
  33: (11,8,5,5,4)
  33: (11,7,6,6,3)
  33: (11,7,3,3,3,3,3)
  33: (11,5,5,4,4,4)
  33: (10,9,8,3,3)
  33: (10,8,6,6,3)
  33: (10,8,3,3,3,3,3)

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326015, A326017, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Range[Max@@#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]

A365922 Number of non-subset-sums of strict integer partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 8, 11, 18, 25, 38, 51, 70, 93, 122, 159, 206, 263, 328, 420, 514, 645, 776, 967, 1154, 1413, 1686, 2042, 2414, 2890, 3394, 4062, 4732, 5598, 6494, 7652, 8836, 10329, 11884, 13833, 15830, 18376, 20936, 24131, 27476, 31547, 35780, 40966, 46292, 52737

Offset: 1

Gus Wiseman, Sep 23 2023

nonn

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The a(6) = 11 ways, showing each strict partition and its non-subset-sums:
    (6): 1,2,3,4,5
   (51): 2,3,4
   (42): 1,3,5
  (321):

The complement (positive subset-sums) is A284640, non-strict A276024.
Weighted row sums of A365545, non-strict A365923.
Row sums of A365663, non-strict A046663.
The non-strict version is A365918.
The zero-full complement (subset-sums) is A365925, non-strict A304792.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A325799, A364272, A365658, A365919, A365921.

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

A262671 Number of pointed multiset partitions of normal pointed multisets of weight n.

Original entry on oeis.org

1, 6, 42, 335, 2956, 28468, 296540

Offset: 1

Gus Wiseman, Sep 26 2015

A pointed multiset k[1...k...n] with point k is normal if its entries [1...k...n] span an initial interval of positive integers. Pointed multiset partitions are triangles (or compositions) in the multiorder of pointed multisets.

			The a(2) = 6 pointed multiset partitions are:
1[1[11]],1[1[1]1[1]],
1[1[12]],1[1[1]2[2]],
2[2[12]],2[1[1]2[2]].
The a(3) = 42 pointed multiset partitions are:
1[1[111]],1[1[1]1[11]],1[1[11]1[1]],1[1[1]1[1]1[1]],
1[1[122]],1[1[1]2[22]],1[1[12]2[2]],1[1[1]2[2]2[2]],
2[2[122]],2[1[1]2[22]],2[1[12]2[2]],2[2[2]2[12]],2[2[12]2[2]],2[1[1]2[2]2[2]],
1[1[112]],1[1[1]1[12]],1[1[1]2[12]],1[1[11]2[2]],1[1[12]1[1]],1[1[1]1[1]2[2]],
2[2[112]],2[1[1]2[12]],2[1[11]2[2]],2[1[1]1[1]2[2]],
1[1[123]],1[1[1]2[23]],1[1[1]3[23]],1[1[12]3[3]],1[1[13]2[2]],1[1[1]2[2]3[3]],
2[2[123]],2[1[1]2[23]],2[1[13]2[2]],2[2[2]3[13]],2[2[12]3[3]],2[1[1]2[2]3[3]],
3[3[123]],3[1[1]3[23]],3[1[12]3[3]],3[2[2]3[13]],3[2[12]3[3]],3[1[1]2[2]3[3]].

Gus Wiseman, Comcategories and Multiorders

Cf. A185298, A080108, A276024.

ReplaceListRepeated[forms_List, rerules_List] :=
Union[Flatten[
   FixedPointList[
    Function[preforms,
     Union[Flatten[ReplaceList[#, rerules] & /@ preforms, 1]]],
    forms], 1]]
pointedPartitions[JIX[r_, b_List?OrderedQ]] /; MemberQ[b, r] :=
  Cases[ReplaceListRepeated[{Z[Y[JIX[r, {r}]],
      Y @@ DeleteCases[b, r, 1, 1]]}, {Z[Y[sof___, JIX[w_, t_]],
        Y[for___, x_, aft___]] /; OrderedQ[{w, x}] :>
      Z[Y[sof, JIX[w, t], JIX[x, {x}]], Y[for, aft]],
     Z[Y[JIX[w_, t_], soa___], Y[for___, x_, aft___]] /;
       OrderedQ[{x, w}] :>
      Z[Y[JIX[x, {x}], JIX[w, t], soa], Y[for, aft]],
     Z[Y[sof___, JIX[w_, {tof__}]], Y[for___, x_, aft___]] :>
      Z[Y[sof, JIX[w, Sort[{tof, x}]]], Y[for, aft]],
     Z[Y[JIX[w_, {tof__}], soa___], Y[for___, x_, aft___]] :>
      Z[Y[JIX[w, Sort[{tof, x}]], soa], Y[for, aft]]}],
   Z[Y[pts__], Y[]] :> JIX[r, {pts}]];
allnormpms[n_Integer] :=
  Join @@ Function[s,
     JIX[#, Array[Count[s, y_ /; y <= #] + 1 &, n]] & /@
      Range[Length[s] + 1]] /@ Subsets[Range[n - 1] + 1];
Join @@ pointedPartitions /@ allnormpms[3] /.
JIX -> Apply(* to construct the example *)
Array[Plus @@ (Length[pointedPartitions[#]] & /@
     allnormpms[#]) &, 7](* to compute the sequence *)

A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 0, 1, 3, 4, 2, 0, 0, 1, 0, 1, 4, 3, 3, 0, 0, 0, 1, 0, 1, 4, 7, 2, 2, 0, 0, 0, 1, 0, 1, 5, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 5, 10, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 6, 9, 5, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, May 15 2019

A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  0  1
  0  1  2  2  0  1
  0  1  3  2  0  0  1
  0  1  3  4  2  0  0  1
  0  1  4  3  3  0  0  0  1
  0  1  4  7  2  2  0  0  0  1
  0  1  5  6  4  2  0  0  0  0  1
  0  1  5 10  6  4  2  0  0  0  0  1
  0  1  6  9  5  1  2  0  0  0  0  0  1
  0  1  6 14 10  5  2  2  0  0  0  0  0  1
  0  1  7 13 11  3  3  2  0  0  0  0  0  0  1
  0  1  7 19 16  7  3  2  2  0  0  0  0  0  0  1
Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):
   (C)  (66)   (444)   (3333)  (81111)  (222222)  (111111111111)
        (75)   (543)   (5511)           (711111)
        (84)   (552)   (7221)
        (93)   (732)   (7311)
        (A2)   (741)   (9111)
        (B1)   (822)
               (831)
               (921)
               (A11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010

Row sums are A000041.
Column k = 2 is A004526.
Column k = 3 is A325690.
Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.

Table[Length[Select[IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15},{k,0,n}]

A325679 Number of compositions of n such that every restriction to a circular subinterval has a different sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 5, 13, 13, 27, 21, 41, 41, 77, 63, 143, 129, 241, 203, 385, 347, 617, 491, 947, 835, 1445, 1185, 2511, 1991, 3585, 2915, 5411, 4569, 8063, 6321, 11131, 10133, 16465, 13207, 23817, 20133, 33929, 26663, 48357, 41363, 69605, 54363, 95727, 81183, 132257, 106581

Offset: 0

Gus Wiseman, May 13 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.
A circular subinterval is a sequence of consecutive indices where the first and last indices are also considered consecutive.
For n > 0, a(n) is the number of subsets of Z_n which contain 0 and such that every ordered pair of distinct elements has a different difference (modulo n). The elements of a subset correspond with the partial sums of a composition. For example, when n = 8 the subset {0,2,7} corresponds with the composition (251). - Andrew Howroyd, Mar 24 2025

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)
            (12)  (13)  (14)  (15)  (16)   (17)
            (21)  (31)  (23)  (24)  (25)   (26)
                        (32)  (42)  (34)   (35)
                        (41)  (51)  (43)   (53)
                                    (52)   (62)
                                    (61)   (71)
                                    (124)  (125)
                                    (142)  (152)
                                    (214)  (215)
                                    (241)  (251)
                                    (412)  (512)
                                    (421)  (521)

Andrew Howroyd, Table of n, a(n) for n = 0..80

Cf. A000079, A008965, A108917, A143823, A169942, A276024.

Cf. A325545, A325676, A325677, A325678, A325680, A325681, A325687, A382399.

suball[q_]:=Join[Take[q,#]&/@Select[Tuples[Range[Length[q]],2],OrderedQ],Drop[q,#]&/@Select[Tuples[Range[2,Length[q]-1],2],OrderedQ]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@suball[#]&]],{n,0,15}]

a(n)={
   my(recurse(k,b,w)=
      if(k >= n, 1,
         b+=1<Andrew Howroyd, Mar 24 2025

a(21) onwards from Andrew Howroyd, Mar 24 2025

A365662 Number of ordered pairs of disjoint strict integer partitions of n.

Original entry on oeis.org

1, 0, 0, 2, 2, 6, 8, 14, 18, 32, 42, 66, 92, 136, 190, 280, 374, 532, 744, 1014, 1366, 1896, 2512, 3384, 4526, 6006, 7910, 10496, 13648, 17842, 23338, 30116, 38826, 50256, 64298, 82258, 105156, 133480, 169392, 214778, 270620, 340554, 428772, 536302, 670522

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of ways to first choose a strict partition of 2n, then a subset of it summing to n.

			The a(0) = 1 through a(7) = 14 pairs:
  ()()  .  .  (21)(3)  (31)(4)  (32)(5)   (42)(6)   (43)(7)
              (3)(21)  (4)(31)  (41)(5)   (51)(6)   (52)(7)
                                (5)(32)   (6)(42)   (61)(7)
                                (5)(41)   (6)(51)   (7)(43)
                                (32)(41)  (321)(6)  (7)(52)
                                (41)(32)  (42)(51)  (7)(61)
                                          (51)(42)  (421)(7)
                                          (6)(321)  (43)(52)
                                                    (43)(61)
                                                    (52)(43)
                                                    (52)(61)
                                                    (61)(43)
                                                    (61)(52)
                                                    (7)(421)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

For subsets instead of partitions we have A000244, non-disjoint A000302.
If the partitions can have different sums we get A032302.
The non-strict version is A054440, non-disjoint A001255.
The unordered version is A108796, non-strict A260669.
A000041 counts integer partitions, strict A000009.
A000124 counts distinct possible sums of subsets of {1..n}.
A000712 counts distinct submultisets of partitions.
A002219 and A237258 count partitions of 2n including a partition of n.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A006827, A046663, A064914, A122768, A260664, A365661, A365663.

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2], Intersection@@#=={}&]], {n,0,15}]
Table[SeriesCoefficient[Product[(1 + x^k + y^k), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Apr 24 2025 *)

a(n) = 2*A108796(n) for n > 1.

a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k). - Ilya Gutkovskiy, Apr 24 2025

A301856 Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 14, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 27, 4, 4, 4, 36, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 62, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 47, 4, 17, 1, 12, 4, 17, 1, 90, 1, 4, 12, 12, 4, 17

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

For a finite multiset p of positive integers greater than 1 with product n, a pair (t > 1, p) is defined to be a subset-product if there exists a nonempty submultiset of p with product t.

			The a(12) = 12 subset-products:
12<=(2*2*3), 6<=(2*2*3), 4<=(2*2*3), 3<=(2*2*3), 2<=(2*2*3),
12<=(2*6),   6<=(2*6),   4<=(3*4),   3<=(3*4),   2<=(2*6),
12<=(3*4),
12<=(12).
The a(16) = 14 subset-products:
16<=(16),
16<=(4*4),
16<=(2*8),     8<=(2*8),     4<=(4*4),     2<=(2*8),
16<=(2*2*4),   8<=(2*2*4),   4<=(2*2*4),   2<=(2*2*4),
16<=(2*2*2*2), 8<=(2*2*2*2), 4<=(2*2*2*2), 2<=(2*2*2*2).

Cf. A001055, A000712, A045778, A108917, A162247, A276024, A281116, A284640, A292886, A301829, A301830.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[Times@@@Rest[Subsets[f]]]],{f,facs[n]}],{n,100}]

cp's OEIS Frontend

A301988 Nonprime Heinz numbers of integer partitions whose product is equal to their sum.

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A305611 Number of distinct positive subset-sums of the multiset of prime factors of n.

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A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

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A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

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A365922 Number of non-subset-sums of strict integer partitions of n.

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A262671 Number of pointed multiset partitions of normal pointed multisets of weight n.

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A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

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A325679 Number of compositions of n such that every restriction to a circular subinterval has a different sum.

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