A284640 -id:A284640 - cp's OEIS frontend

A299702 Heinz numbers of knapsack partitions.

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, 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, 77, 78

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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

Cf. A056239, A108917, A112798, A275972, A276024, A284640, A296150, A299701, A299729.

filter:= proc(n) local F,t,S,i,r;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}: r:= 1;
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
   r:= r*(t[2]+1);
   if nops(S) <> r then return false fi
  od;
  true
end proc:
select(filter, [$1..100]); # Robert Israel, Oct 30 2024

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Plus@@@Union[Rest@Subsets[primeMS[#]]]&]

A301987 Heinz numbers of integer partitions whose product is equal to their sum.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 200, 211, 223, 227, 229, 233, 239, 241, 251

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: (1), (2), (3), (4), (2 2), (5), (6), (7), (8), (9), (10), (1 2 3), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (1 1 2 4), (24), (25), (26), (27), (28), (1 1 2 2 2), (29), (30).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001055, A002865, A003963, A056239, A276024, A284640, A296150, A299701, A301957, A301988.

q:= n-> (l-> mul(i, i=l)=add(i, i=l))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..300])[];  # Alois P. Heinz, Mar 27 2019

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[300],Total[primeMS[#]]===Times@@primeMS[#]&]

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer n is a subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Position of first appearance of n appears to be A259941(n-1) = least Heinz number of a complete partition of n-1. - Gus Wiseman, Nov 16 2023

			The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.
Positions of first appearances are A259941.
The triangle for this rank statistic is A365658.
The semi version is A366739, sum A366738, strict A366741.
Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.

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

a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).

Comment corrected by Gus Wiseman, Aug 09 2024

A304792 Number of subset-sums of integer partitions of n.

Original entry on oeis.org

1, 2, 5, 10, 19, 34, 58, 96, 152, 240, 361, 548, 795, 1164, 1647, 2354, 3243, 4534, 6150, 8420, 11240, 15156, 19938, 26514, 34513, 45260, 58298, 75704, 96515, 124064, 157072, 199894, 251097, 317278, 395625, 496184, 615229, 765836, 944045, 1168792, 1432439

Offset: 0

Gus Wiseman, May 18 2018

nonn

For a multiset p of positive integers summing to n, a pair (t,p) is defined to be a subset sum if there exists a submultiset of p summing to t. This sequence is dominated by A122768 + A000041 (number of submultisets of integer partitions of n).

			The a(4)=19 subset sums are (0,4), (4,4), (0,31), (1,31), (3,31), (4,31), (0,22), (2,22), (4,22), (0,211), (1,211), (2,211), (3,211), (4,211), (0,1111), (1,1111), (2,1111), (3,1111), (4,1111).

Robert Price, Table of n, a(n) for n = 0..70

Cf. A000041, A108917, A122768, A276024, A284640, A299701, A301856, A301934, A301979, A304793.

b:= proc(n, i, s) option remember; `if`(n=0, nops(s),
     `if`(i<1, 0, b(n, i-1, s)+b(n-i, min(n-i, i),
      map(x-> [x, x+i][], s))))
    end:
a:= n-> b(n$2, {0}):
seq(a(n), n=0..40);  # Alois P. Heinz, May 18 2018

Table[Total[Length[Union[Total/@Subsets[#]]]&/@IntegerPartitions[n]],{n,15}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0, Length[s],
     If[i < 1, 0, b[n, i - 1, s] + b[n - i, Min[n - i, i],
     {#, # + i}& /@ s // Flatten // Union]]];
a[n_] := b[n, n, {0}];
a /@ Range[0, 40] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A304792_T(n,i,s,l):
    if n==0: return l
    if i<1: return 0
    return A304792_T(n,i-1,s,l)+A304792_T(n-i,min(n-i,i),(t:=tuple(sorted(set(s+tuple(x+i for x in s))))),len(t))
def A304792(n): return A304792_T(n,n,(0,),1) # Chai Wah Wu, Sep 25 2023, after Alois P. Heinz

a(n) = A276024(n) + A000041(n).

A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

Original entry on oeis.org

1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304

Offset: 1

N. J. A. Sloane

			Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - _N. J. A. Sloane_, Jun 03 2012
From _Gus Wiseman_, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
  (11)  (22)    (33)      (44)
        (211)   (321)     (422)
        (1111)  (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.

Column m=2 of A213086.
Bisection of A276107.
Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.
The strict version is A237258, ranked by A357854.
Ranked by A357976 = positions of nonzero terms in A357879.
A122768 counts distinct submultisets of partitions.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A108917, A235130, A237194, A300061.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,
      b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,
      max(x, n-i-x)), `if`(xn, NULL, max(x-i, n-x))}[], s)))))
    end:
a:= n-> b(2*n, n, {n}):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

Mathematica

b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *) primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s]; Table[Length[Select[IntegerPartitions[2n],MemberQ[Total/@subptns[#],n]&]],{n,10}] (* Gus Wiseman, Oct 27 2022 *)

Python

from itertools import combinations_with_replacement from sympy.utilities.iterables import partitions def A002219(n): return len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023

Formula

See A213074 for Metropolis and Stein's formulas.

a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n).

a(n) = A276107(2*n). - Max Alekseyev, Oct 17 2022

Extensions

Better description from Vladeta Jovovic, Mar 06 2000

More terms from Christian G. Bower, Oct 12 2001

Edited by N. J. A. Sloane, Jun 03 2012

More terms from Alois P. Heinz, Jul 10 2012

A316313 Number of integer partitions of n such that every distinct submultiset has a different average.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 18, 22, 28, 26, 34, 37, 44, 50, 61, 53, 77, 82, 93, 89, 120, 120, 149, 138, 174, 180, 222, 193, 257, 262, 305, 281, 367, 359, 424, 398, 487, 507, 590, 526, 662, 666, 782, 729, 894, 892, 995, 987, 1154, 1188, 1370

Offset: 1

Gus Wiseman, Jun 29 2018

nonn

Note that such a partition is necessarily strict.

			The a(8) = 6 integer partitions are (8), (71), (62), (53), (521), (431).

Cf. A000009, A108917, A275972, A276024, A284640, A299702, A301899, A301900, A316271, A316314.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Mean/@Union[Subsets[#]]&]],{n,20}]

A237258 Number of strict partitions of 2n that include a partition of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 32, 43, 63, 84, 122, 158, 220, 293, 393, 511, 685, 881, 1156, 1485, 1925, 2445, 3147, 3952, 5019, 6323, 7924, 9862, 12336, 15259, 18900, 23294, 28646, 35091, 42985, 52341, 63694, 77336, 93588, 112973, 136367, 163874, 196638

Offset: 0

Clark Kimberling, Feb 05 2014

nonn

A strict partition is a partition into distinct parts.

			a(5) counts these partitions of 10: [5,4,1], [5,3,2], [4,3,2,1].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Cf. A000009, A237194, A235130.
The non-strict version is A002219, ranked by A357976.
These partitions are ranked by A357854.
A000712 counts distinct submultisets of partitions, strict A032302.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A006827, A064914, A108917, A276107, A300061, A357879.

z = 24; Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &,  Select[IntegerPartitions[2 nn], # == DeleteDuplicates[#] &]]; Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], nn]] >= 1 &]]], {nn, z}] (* Peter J. C. Moses, Feb 04 2014 *)

a(n) = A237194(2n,n).

a(31)-a(47) from Alois P. Heinz, Feb 07 2014

A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 18 2018

nonn

A positive integer n is a positive subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
a(n) <= A000005(n).
One less than the number of distinct values obtained when A056239 is applied to all divisors of n. - Antti Karttunen, Jul 01 2018

			The positive subset-sums of (4,3,1) are {1, 3, 4, 5, 7, 8} so a(70) = 6.
The positive subset-sums of (5,1,1,1) are {1, 2, 3, 5, 6, 7, 8} so a(88) = 7.

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

Cf. A056239, A122768, A276024, A284640, A296150, A299701, A299702, A301855, A301935, A301957, A304792, A304795, A305611.

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

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A304793(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v056239[d]), mapput(m,s,s); k++)); (k-1); }; \\ Antti Karttunen, Jul 01 2018

More terms from Antti Karttunen, Jul 01 2018

A317142 Number of refinement-ordered pairs of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 12, 16, 24, 37, 47, 68, 90, 123, 180, 228, 307, 408, 540, 694, 970, 1207, 1598, 2048, 2669, 3357, 4382, 5599, 7109, 8990, 11428, 14330, 18144, 22652, 28343, 35746, 44269, 55094, 68384, 84780, 104477, 129360, 158682, 195323, 240177, 293704

Offset: 0

Gus Wiseman, Jul 22 2018

nonn

If x and y are strict partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.

This sequence is dominated by A294617 (set partitions of strict partitions).

			The a(9) = 24 refinement-ordered pairs:
    (9)<=(9)
  (5,4)<=(9)   (5,4)<=(5,4)
  (6,3)<=(9)   (6,3)<=(6,3)
  (7,2)<=(9)   (7,2)<=(7,2)
  (8,1)<=(9)   (8,1)<=(8,1)
(4,3,2)<=(9) (4,3,2)<=(5,4) (4,3,2)<=(6,3) (4,3,2)<=(7,2) (4,3,2)<=(4,3,2)
(5,3,1)<=(9) (5,3,1)<=(5,4) (5,3,1)<=(6,3) (5,3,1)<=(8,1) (5,3,1)<=(5,3,1)
(6,2,1)<=(9) (6,2,1)<=(6,3) (6,2,1)<=(7,2) (6,2,1)<=(8,1) (6,2,1)<=(6,2,1)

Robert Price, Table of n, a(n) for n = 0..70

Cf. A002846, A108917, A122768, A213427, A265947, A284640, A294617, A299201, A300383, A317141.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Union[Select[Sort/@Map[Total,mps[ptn],{2}],UnsameQ@@#&]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A299729 Heinz numbers of non-knapsack partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer partition is non-knapsack if there exist two different submultisets with the same sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			12 is the Heinz number of (2,1,1) which is not knapsack because 2 = 1 + 1.

Cf. A056239, A108917, A112798, A275972, A276024, A284640, A296150, A299701, A299702.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@Plus@@@Union[Rest@Subsets[primeMS[#]]]&]

cp's OEIS Frontend

A299702 Heinz numbers of knapsack partitions.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A301987 Heinz numbers of integer partitions whose product is equal to their sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A304792 Number of subset-sums of integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A316313 Number of integer partitions of n such that every distinct submultiset has a different average.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A237258 Number of strict partitions of 2n that include a partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica