A316314 -id:A316314 - cp's OEIS frontend

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

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}]

A316402 Number of strict non-knapsack integer partitions of n, meaning not every subset has a different sum.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 8, 6, 12, 10, 20, 16, 29, 25, 44, 39, 61, 60, 91, 84, 125, 126, 180, 179, 242, 247, 336, 347, 444, 472, 606, 628, 796, 844, 1053, 1109, 1363, 1452, 1779, 1885, 2272, 2431, 2931, 3104, 3706, 3972, 4711, 5042, 5909, 6334

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

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

Cf. A000009, A108917, A275972, A299702, A301899, A301900, A316271, A316314, A316399, A316400.

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

a(n) = A000009(n) - A275972(n).

A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

Note that such a Heinz number is necessarily squarefree, as such a partition is necessarily strict.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
First differs from A301899 at a(43) = 70, because (4,3,1) is not knapsack but every submultiset has a different average.

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).

Cf. A000009, A056239, A122768, A237984, A275972, A299702, A301899, A316313, A316314, A316465.

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

A316556 Number of distinct LCMs 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, 4, 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, 4, 1, 2, 2, 1, 2, 4, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 3, 3, 1, 2, 1, 2, 1, 3, 3, 2, 2, 2, 1, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 4, 1, 2, 5

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 A290103 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 25 2018

			462 is the Heinz number of (5,4,2,1) which has possible LCMs of nonempty submultisets {1,2,4,5,10,20} so a(462) = 6.

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

Cf. A056239, A074761, A108917, A122768, A275972, A290103, A296150, A301957, A316313, A316314, A316430, A316555, A316557.

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

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

A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A316556(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A290103(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

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

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).

A316400 Number of strict integer partitions of n that are knapsack (every subset has a different sum) but not every subset has a different average.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 4, 1, 2, 4, 6, 4, 13, 6, 13, 17, 15, 12, 31, 26, 27, 23, 36, 41, 56, 39, 47, 74, 71, 55, 101, 94, 110, 97, 145, 148, 189, 142, 214, 232, 280, 206, 362, 332, 414, 347, 504, 469, 658, 492, 726, 697, 867, 687, 1100, 933

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

			The a(21) = 13 partitions:
(8,7,6), (9,7,5), (10,7,4), (11,7,3), (12,7,2), (13,7,1),
(7,6,5,3), (8,6,4,3), (9,7,4,1), (10,6,3,2), (11,6,3,1), (12,4,3,2), (12,5,3,1).

Cf. A000009, A108917, A275972, A299702, A301899, A301900, A316271, A316313, A316314, A316399, A316401.

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

A316364 Number of factorizations of n into factors > 1 such that every distinct submultiset of the factors has a different average.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 30 2018

nonn

Note that such a factorization is necessarily strict.

			The a(80) = 6 factorizations are (80), (10*8), (16*5), (20*4), (40*2), (10*4*2).

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

Cf. A001055, A108917, A275972, A276024, A284640, A292886, A293627, A294150, A316313, A316314, A316365.

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],UnsameQ@@Mean/@Union[Subsets[#]]&]],{n,50}]

choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };
hasdupavgs(v) = { my(avgs=Map(), k); for(i=1,(2^(#v))-1,k = (vecsum(choosebybits(v,i))/hammingweight(i)); if(mapisdefined(avgs,k),return(i),mapput(avgs,k,i))); (0); };
A316364(n, m=n, facs=List([])) = if(1==n, (0==hasdupavgs(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A316364(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Sep 21 2018

More terms from Antti Karttunen, Sep 21 2018

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

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A316402 Number of strict non-knapsack integer partitions of n, meaning not every subset has a different sum.

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A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

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A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

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

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

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