A316314 -id:A316314 - cp's OEIS frontend

A316365 Number of factorizations of n into factors > 1 such that every distinct subset of the factors has a different sum.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 6, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 7, 2, 2, 1, 10, 1, 2, 4, 9, 2, 5, 1, 4, 2, 4, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 9, 2, 2, 2, 7, 1, 10, 2, 4, 2, 2, 2, 15, 1, 4, 4, 9, 1, 5, 1, 7, 5

Offset: 1

Gus Wiseman, Jun 30 2018

nonn

Also the number of factorizations of n into factors > 1 which form a knapsack partition.

			The a(24) = 7 factorizations are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24).
The a(54) = 6 factorizations are (2*3*3*3), (2*3*9), (2*27), (3*18), (6*9), (54).

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A108917, A275972, A292886, A293627, A294150, A299702, A316313, A316314, A316364.

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@@Total/@Union[Subsets[#]]&]],{n,100}]

primeprodbybits(v,b) = { my(m=1,i=1); while(b>0,if(b%2, m *= prime(v[i])); i++; b >>= 1); (m); };
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };
all_distinct_subsets_have_different_sums(v) = { my(m=Map(),s,pp); for(i=0,(2^#v)-1, pp = primeprodbybits(v,i); s = sumbybits(v,i); if(mapisdefined(m,s), if(mapget(m,s)!=pp, return(0)), mapput(m,s,pp))); (1); };
A316365(n, m=n, facs=List([])) = if(1==n, all_distinct_subsets_have_different_sums(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A316365(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

Although the average of an empty set is technically indeterminate, we consider it to be distinct from the other subset-averages.

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

			The a(60) = 9 distinct subset-averages of (3,2,1,1) are 0/0, 1, 4/3, 3/2, 5/3, 7/4, 2, 5/2, 3.

Cf. A032302, A056239, A108917, A275972, A276024, A296150, A299701, A301899, A301957, A316313.

One more than A316314.

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

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A316398(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = A056239(d)/bigomega(d)), mapput(m,s,s); k++)); (1+k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316314(n) + 1.

More terms from Antti Karttunen, Sep 23 2018

A319335 Numerator of the average of the averages of all integer partitions of n.

Original entry on oeis.org

1, 3, 11, 31, 187, 131, 247, 1993, 4463, 3635, 395077, 24441, 81149, 10414421, 12868591, 10764151, 61170133, 419426561, 353495183, 3429826973, 29219934899, 5110021867, 142319532929, 606916707064, 87086496509, 4426308633083, 15954910019953, 38414031851849

Offset: 1

Gus Wiseman, Sep 17 2018

			The sequence of average averages begins: 1, 3/2, 11/6, 31/15, 187/84, 131/55, 247/100, 1993/770, 4463/1680, 3635/1323.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Denominators are in A319336.

Cf. A056239, A067538, A237984, A296150, A316313, A316314, A316440.

Table[Numerator[Mean[Mean/@IntegerPartitions[n]]],{n,20}]

seq(n)={[numerator(poldegree(p)*subst(intformal(p/y)/p, y, 1)) | p <- Vec(-1+1/prod(k=1, n, 1 - x^k*y + O(x*x^n)))]} \\ Andrew Howroyd, Sep 19 2018

A319336 Denominator of the average of the averages of all integer partitions of n.

Original entry on oeis.org

1, 2, 6, 15, 84, 55, 100, 770, 1680, 1323, 141120, 8470, 27720, 3474900, 4228224, 3468465, 19459440, 131030900, 109156320, 1042578108, 8779605120, 1514663280, 41736380400, 175685635125, 24960905112, 1254125149200, 4476730258000, 10664476594200, 73326164511600

Offset: 1

Gus Wiseman, Sep 17 2018

			The sequence of average averages begins: 1, 3/2, 11/6, 31/15, 187/84, 131/55, 247/100, 1993/770, 4463/1680, 3635/1323.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Numerators are in A319335.

Cf. A056239, A067538, A237984, A296150, A316313, A316314, A316440.

Table[Denominator[Mean[Mean/@IntegerPartitions[n]]],{n,20}]

seq(n)={[denominator(poldegree(p)*subst(intformal(p/y)/p, y, 1)) | p <- Vec(-1+1/prod(k=1, n, 1 - x^k*y + O(x*x^n)))]} \\ Andrew Howroyd, Sep 19 2018

A316362 Heinz numbers of strict integer partitions such that not every distinct subset has a different average.

Original entry on oeis.org

30, 105, 110, 210, 238, 273, 330, 385, 390, 462, 506, 510, 546, 570, 627, 690, 714, 770, 806, 858, 870, 910, 930, 935, 966, 1001, 1110, 1131, 1155, 1190, 1230, 1254, 1290, 1326, 1330, 1365, 1394, 1410, 1430, 1482, 1495, 1518, 1590, 1729, 1770, 1785, 1786, 1794

Offset: 1

Gus Wiseman, Jun 30 2018

nonn

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

			462 is the Heinz number of (5,4,2,1), and the subsets {1,5}, and {2,4} have the same average, so 462 belongs to the sequence.

Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299702, A301899, A316313, A316314, A316361.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[3000],SquareFreeQ[#]&&!UnsameQ@@Mean/@Union[Subsets[primeMS[#]]]&]

A316401 Number of strict integer partitions of n that are not knapsack (not every subset has a different sum) but every subset has a different average.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 3, 1, 3, 2, 4, 2, 6, 6, 11, 9, 9, 10, 20, 16, 18, 17, 27, 24, 31, 29, 43, 31, 43, 40, 59, 52, 58, 61, 83, 68, 93, 80, 124, 99, 120, 109, 145, 151, 185, 160, 232, 163, 257, 229, 314, 280, 286, 310, 427, 385, 513, 333, 596

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

			The a(22) = 11 partitions:
(11,6,5), (11,7,4), (11,8,3), (11,9,2), (11,10,1),
(11,5,4,2), (11,6,3,2), (11,8,2,1), (12,5,3,2), (12,5,4,1), (14,4,3,1).

Cf. A000009, A108917, A275972, A299702, A301899, A301900, A316271, A316313, A316314, A316399, A316400.

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

A316465 Heinz numbers of integer partitions such that every nonempty submultiset has an integer average.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 79, 81, 82, 83, 85, 87, 89, 91, 94, 97, 101, 103, 107, 109, 110, 111, 113, 115, 118, 121, 125, 127, 128

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).

Supersequence of A000961. - David A. Corneth, Jul 06 2018

			Sequence of partitions begins (), (1), (2), (1,1), (3), (4), (1,1,1), (2,2), (3,1), (5), (6), (1,1,1,1), (7), (8), (4,2), (5,1), (9), (3,3), (2,2,2).

Cf. A056239, A067538, A122768, A237984, A296150, A316313, A316314, A316440, A316557.

Select[Range[100],And@@IntegerQ/@Mean/@Union[Rest[Subsets[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]&]

cp's OEIS Frontend

A316365 Number of factorizations of n into factors > 1 such that every distinct subset of the factors has a different sum.

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A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

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A319335 Numerator of the average of the averages of all integer partitions of n.

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A319336 Denominator of the average of the averages of all integer partitions of n.

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A316362 Heinz numbers of strict integer partitions such that not every distinct subset has a different average.

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A316401 Number of strict integer partitions of n that are not knapsack (not every subset has a different sum) but every subset has a different average.

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A316465 Heinz numbers of integer partitions such that every nonempty submultiset has an integer average.

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