A316313 -id:A316313 - cp's OEIS frontend

A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

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

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of product-sum knapsack partitions begins:
   0: ()
   1:
   2: (2)
   3: (3)
   4: (4)
   5: (5) (3,2)
   6: (6) (4,2) (3,3)
   7: (7) (5,2) (4,3)
   8: (8) (6,2) (5,3) (4,4)
   9: (9) (7,2) (6,3) (5,4)
  10: (10) (8,2) (7,3) (6,4) (5,5) (4,3,3)
  11: (11) (9,2) (8,3) (7,4) (6,5) (5,4,2) (5,3,3) (4,4,3)
  12: (12) (10,2) (9,3) (8,4) (7,5) (7,3,2) (6,6) (4,4,4)
A complete list of all products of sums of multiset partitions of submultisets of (4,3,3) is:
           () = 1
          (3) = 3
          (4) = 4
        (3+3) = 6
        (3+4) = 7
      (3+3+4) = 10
      (3)*(3) = 9
      (3)*(4) = 12
    (3)*(3+4) = 21
    (4)*(3+3) = 24
  (3)*(3)*(4) = 36
These are all distinct, so (4,3,3) is a product-sum knapsack partition of 10.

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320053, A320054, A320055, A320056, A320057, A320058.

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]]]];
rrsuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Times,Apply[Plus,Union@@mps/@Union[Subsets[q]],{2}],{1}]]];
Table[Length[rrsuks[n]],{n,12}]

A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 6, 8, 9, 12, 14

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of spanning sum-product knapsack partitions begins:
  0: ()
  1: (1)
  2: (2) (1,1)
  3: (3) (2,1) (1,1,1)
  4: (4) (3,1)
  5: (5) (4,1) (3,2)
  6: (6) (5,1) (4,2) (3,3)
  7: (7) (6,1) (5,2) (4,3) (3,3,1)
  8: (8) (7,1) (6,2) (5,3) (4,4) (3,3,2)
  9: (9) (8,1) (7,2) (6,3) (5,4) (4,4,1) (4,3,2) (3,3,3)
A complete list of all sums of products covering the parts of (3,3,3,2) is:
        (2*3*3*3) = 54
      (2)+(3*3*3) = 29
      (3)+(2*3*3) = 21
      (2*3)+(3*3) = 15
    (2)+(3)+(3*3) = 14
    (3)+(3)+(2*3) = 12
  (2)+(3)+(3)+(3) = 11
These are all distinct, so (3,3,3,2) is a spanning sum-product knapsack partition of 11.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320052, A320054, A320055, A320056, A320057, A320058.

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]]]];
rtuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Plus,Apply[Times,mps[q],{2}],{1}]]];
Table[Length[rtuks[n]],{n,8}]

A320054 Number of spanning product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums the parts of a multiset partition of y is distinct.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 5, 8, 10, 12, 16, 17, 25

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of spanning product-sum knapsack partitions begins
0: ()
1: (1)
2: (2) (1,1)
3: (3) (2,1) (1,1,1)
4: (4) (3,1)
5: (5) (4,1) (3,2) (3,1,1)
6: (6) (5,1) (4,2) (4,1,1) (3,3)
7: (7) (6,1) (5,2) (5,1,1) (4,3) (4,2,1) (4,1,1,1) (3,3,1)
8: (8) (7,1) (6,2) (6,1,1) (5,3) (5,2,1) (5,1,1,1) (4,4) (4,3,1) (3,3,2)
9: (9) (8,1) (7,2) (7,1,1) (6,3) (6,2,1) (6,1,1,1) (5,4) (5,3,1) (4,4,1) (4,3,2) (3,3,3)
A complete list of all products of sums covering the parts of (4,1,1,1) is:
        (1+1+1+4) = 7
      (1)*(1+1+4) = 6
      (4)*(1+1+1) = 12
      (1+1)*(1+4) = 10
    (1)*(1)*(1+4) = 5
    (1)*(4)*(1+1) = 8
  (1)*(1)*(1)*(4) = 4
These are all distinct, so (4,1,1,1) is a spanning product-sum knapsack partition of 7.
A complete list of all products of sums covering the parts of (5,3,1,1) is:
        (1+1+3+5) = 10
      (1)*(1+3+5) = 9
      (3)*(1+1+5) = 21
      (5)*(1+1+3) = 25
      (1+1)*(3+5) = 16
      (1+3)*(1+5) = 24
    (1)*(1)*(3+5) = 8
    (1)*(3)*(1+5) = 18
    (1)*(5)*(1+3) = 20
    (3)*(5)*(1+1) = 30
  (1)*(1)*(3)*(5) = 15
These are all distinct, so (5,3,1,1) is a spanning product-sum knapsack partition of 10.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320052, A320053, A320055, A320056, A320057, A320058.

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]]]];
rsuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Times,Apply[Plus,mps[q],{2}],{1}]]];
Table[Length[rsuks[n]],{n,10}]

A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 1, 0, 1, 1, 0, 0, 6, 0, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 4, 0, 0, 1, 4, 0, 1, 0, 1, 0, 1, 0, 9, 0, 0, 1, 1, 0, 1, 0, 6, 1, 0, 0, 5, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 13, 0, 1, 1, 3, 0, 1, 0, 3, 0, 0, 0, 10

Offset: 1

Gus Wiseman, Jan 24 2023

nonn

			The a(72) = 9 factorizations: (2*2*2*3*3), (2*2*2*9), (2*2*3*6), (2*2*18), (2*3*12), (2*4*9), (2*6*6), (3*3*8), (3*4*6).

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

The version for partitions is A359894, complement A240219.
The complement is counted by A359909, odd-length A359910.
A001055 counts factorizations.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A316313, A326567/A326568, A359897, A359889, A359906, A360005.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Mean[#]!=Median[#]&]],{n,100}]

median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
A359911(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)!=(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359911(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

For n > 1, a(n) = A001055(n) - A359909(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A319320 Number of integer partitions of n such that every distinct submultiset has a different LCM.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 12, 15, 17, 20, 22, 24, 25, 31, 35, 39, 40, 48, 51, 55, 64, 73, 77, 85, 92, 104, 115, 126, 136, 147, 157, 176, 198, 211, 234, 246, 269, 294, 326, 350, 375, 403, 443, 475, 526, 560, 600, 650

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

Note that such partitions are necessarily strict.

			The a(19) = 12 partitions:
  (19),
  (10,9), (11,8), (12,7), (13,6), (14,5), (15,4), (16,3), (17,2),
  (8,6,5), (11,5,3),
  (7,5,4,3).

Cf. A074761, A108917, A275972, A290103, A316313, A316429, A316431, A319315, A319318, A319327.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@LCM@@@Union[Rest[Subsets[#]]]&]],{n,30}]

A316440 Number of integer partitions of n such that every submultiset has an integer average.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 2, 7, 5, 8, 2, 13, 2, 10, 10, 14, 2, 20, 2, 17, 15, 14, 2, 32, 3, 16, 22, 25, 2, 40, 2, 27, 30, 20, 4, 58, 2, 22, 40, 40, 2, 64, 2, 40, 53, 26, 2, 93, 3, 30, 64, 54, 2, 94, 4, 58, 78, 32, 2, 138, 2, 34, 96, 75, 10, 131, 2, 76, 111, 48, 2, 192, 2, 40, 138, 99

Offset: 0

Gus Wiseman, Jul 03 2018

nonn

			The a(12) = 13 partitions:
  (12),
  (6,6), (7,5), (8,4), (9,3), (10,2), (11,1),
  (4,4,4), (6,4,2), (8,2,2),
  (3,3,3,3),
  (2,2,2,2,2,2),
  (1,1,1,1,1,1,1,1,1,1,1,1).

Max Alekseyev, Table of n, a(n) for n = 0..500

Cf. A067538, A074761, A122768, A143773, A237984, A298423, A316313, A316314.

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

For a prime p, a(p) = 2. - Max Alekseyev, Sep 02 2023

a(0) prepended and more terms added by Max Alekseyev, Sep 02 2023

A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

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

Number of distinct values obtained when A289508 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 28 2018

			455 is the Heinz number of (6,4,3) which has possible GCDs of nonempty submultisets {1,2,3,4,6} so a(455) = 5.

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

Cf. A056239, A108917, A122768, A275972, A289508, A289509, A296150, A316313, A316314, A316430, A316556, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[GCD@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A289508(n) = gcd(apply(p->primepi(p),factor(n)[,1]));
A316555(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A289508(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 28 2018

More terms from Antti Karttunen, Sep 28 2018

A316557 Number of distinct integer averages of subsets of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 3, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 3, 2, 1, 3, 1, 3, 3, 1, 2, 4, 1, 4, 2, 4, 1, 2, 1, 2, 2, 2, 2, 5, 1, 3, 1, 3, 1, 4, 3, 2, 3, 4, 1, 3, 3, 3, 2, 3, 2, 2, 1, 3, 3, 3, 1, 4, 1, 2, 3

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

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

			The a(78) = 5 distinct integer averages of subsets of (6,2,1) are {1, 2, 3, 4, 6}.

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

Cf. A056239, A067538, A122768, A237984, A296150, A316313, A316314, A316440, A316555, A316556.

Table[Length[Select[Union[Mean/@Subsets[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],IntegerQ]],{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));
A316557(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&(1==denominator(s = v056239[d]/bigomega(d)))&&!mapisdefined(m,s), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

a(n) <= A316314(n). - Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A319319 Heinz numbers of integer partitions such that every distinct submultiset has a different GCD.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 177

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

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

First differs from A304713 (Heinz numbers of pairwise indivisible partitions) at A304713(58) = 165, which is absent from this sequence because its prime indices are {2,3,5} and GCD(2,3) = GCD(2,3,5) = 1. The first term with more than two prime factors is 17719, which has prime indices {6,10,15}. The first term with more than two prime factors that is absent from A318716 is 296851, which has prime indices {12,20,30}.

			The sequence of partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (3), (4), (5), (6), (3,2), (7), (8), (9), (10), (11), (5,2), (4,3), (12), (13), (14), (15), (7,2), (16), (5,3).

Cf. A056239, A108917, A122768, A275972, A299702, A301899, A301900, A304713, A316313, A319315, A319318, A319327.

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

A316399 Number of strict integer partitions of n such that not every subset has a different average.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 5, 3, 5, 9, 10, 10, 20, 20, 27, 32, 39, 43, 69, 65, 83, 99, 133, 136, 176, 191, 252, 274, 332, 363, 475, 503, 602, 677, 832, 893, 1067, 1186, 1418, 1561, 1797, 2000, 2384, 2602, 2992, 3315, 3853, 4226, 4826, 5383, 6121, 6763

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

			The a(12) = 5 partitions are (5,4,3), (6,4,2), (7,4,1), (5,4,2,1), (6,3,2,1).

Cf. A000009, A108917, A275972, A276024, A284640, A299702, A301899, A301900, A316271, A316313, A316314, A316400, A316402.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!UnsameQ@@Mean/@Union[Subsets[#]]&]],{n,60}]

a(n) = A000009(n) - A316313(n).

cp's OEIS Frontend

A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

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A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

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A320054 Number of spanning product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums the parts of a multiset partition of y is distinct.

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A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

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A319320 Number of integer partitions of n such that every distinct submultiset has a different LCM.

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A316440 Number of integer partitions of n such that every submultiset has an integer average.

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A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

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PARI

Extensions

A316557 Number of distinct integer averages of subsets of the integer partition with Heinz number n.

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PARI