A035470 Number of ways to break {1,2,3,...,n} into sets with equal sums.

1, 1, 2, 2, 2, 2, 6, 12, 11, 2, 80, 166, 2, 665, 2918, 3309, 9296, 23730, 31875, 301030, 422897, 2, 13716867, 71504980, 100664385, 54148591, 880696662, 498017759, 27450476787, 111911522819, 179459955554, 2144502175214, 59115423983, 45837019664552, 375743493787258, 816118711787493, 2, 9492169507922

Offset: 1

Author

Erich Friedman

Keywords

• nonn

Comments

a(n) = 2 <=> |{d|n*(n+1)/2 : d>=n}| = 2. - Alois P. Heinz, Sep 03 2009

Examples

			a(7) = 6 since we have 1234567, 16/25/34/7, 167/2345, 257/1346, 347/1256, 356/1247.
From _Gus Wiseman_, Jul 13 2019: (Start)
The a(6) = 2 through a(9) = 11 set partitions with equal block-sums:
  {123456}      {1234567}        {12345678}        {123456789}
  {16}{25}{34}  {1247}{356}      {12348}{567}      {12345}{69}{78}
                {1256}{347}      {12357}{468}      {1239}{456}{78}
                {1346}{257}      {12456}{378}      {1248}{357}{69}
                {167}{2345}      {1278}{3456}      {1257}{348}{69}
                {16}{25}{34}{7}  {1368}{2457}      {1347}{258}{69}
                                 {1458}{2367}      {1356}{249}{78}
                                 {1467}{2358}      {159}{2346}{78}
                                 {1236}{48}{57}    {159}{267}{348}
                                 {138}{246}{57}    {168}{249}{357}
                                 {156}{237}{48}    {18}{27}{36}{45}{9}
                                 {18}{27}{36}{45}
(End)
		

Links

• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Crossrefs

Cf. A164977, A164978. - Alois P. Heinz, Sep 03 2009

Row sums of A275714.

Row sums of A320438.

Cf. A000110, A007837, A038041, A112956, A275780, A275781, A321455, A326512, A326513, A326518, A326534.

Programs

Maple

with(numtheory): b:= proc() option remember; local i, j, t; `if`(args[1]=0, `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(`if`(args[j] -args[nargs] <0, 0, b(sort([seq(args[i] -`if`(i=j, args[nargs], 0), i=1..nargs-1)])[], args[nargs]-1)), j=1..nargs-1)) end: a:= proc(n) local i, m, x; m:= n*(n+1)/2; 1+ add(b(i$(m/i), n)/(m/i)!, i=[select(x-> x>=n, divisors(m) minus {m})[]]) end: seq(a(n), n=1..25);  # Alois P. Heinz, Sep 03 2009

Mathematica

b[args_List] := b[args] = If[args[[1]] == 0, If[Length[args] == 2, 1, b[Rest[args]]], Sum[If[args[[j]] - args[[-1]] < 0, 0, b[Sort[Join[Table[ args[[i]] - If[i == j, args[[-1]], 0], {i, 1, Length[args]-1}]]], {args[[-1]]-1}]], {j, 1, Length[args]-1}]]; b[a1_List, a2_List] := b[Join[a1, a2]];
a[n_] := a[n] = With[{m = n*(n+1)/2}, 1+Sum[b[Append[Array[i&, m/i], n]] / (m/i)!, {i, Select[Divisors[m] ~Complement~ {m}, # >= n &]}]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Mar 22 2017, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Total/@#&]],{n,0,10}] (* Gus Wiseman, Jul 13 2019 *)

Extensions

More terms from John W. Layman, Mar 18 2002

a(19)-a(33) from Alois P. Heinz, Sep 03 2009

a(34) from Alois P. Heinz, May 24 2015

a(35)-a(38) from Max Alekseyev, Feb 15 2024