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

%I A035470 #38 Feb 27 2025 12:45:13
%S A035470 1,1,2,2,2,2,6,12,11,2,80,166,2,665,2918,3309,9296,23730,31875,301030,
%T A035470 422897,2,13716867,71504980,100664385,54148591,880696662,498017759,
%U A035470 27450476787,111911522819,179459955554,2144502175214,59115423983,45837019664552,375743493787258,816118711787493,2,9492169507922
%N A035470 Number of ways to break {1,2,3,...,n} into sets with equal sums.
%C A035470 a(n) = 2 <=> |{d|n*(n+1)/2 : d>=n}| = 2. - _Alois P. Heinz_, Sep 03 2009
%H A035470 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict</a>.
%e A035470 a(7) = 6 since we have 1234567, 16/25/34/7, 167/2345, 257/1346, 347/1256, 356/1247.
%e A035470 From _Gus Wiseman_, Jul 13 2019: (Start)
%e A035470 The a(6) = 2 through a(9) = 11 set partitions with equal block-sums:
%e A035470   {123456}      {1234567}        {12345678}        {123456789}
%e A035470   {16}{25}{34}  {1247}{356}      {12348}{567}      {12345}{69}{78}
%e A035470                 {1256}{347}      {12357}{468}      {1239}{456}{78}
%e A035470                 {1346}{257}      {12456}{378}      {1248}{357}{69}
%e A035470                 {167}{2345}      {1278}{3456}      {1257}{348}{69}
%e A035470                 {16}{25}{34}{7}  {1368}{2457}      {1347}{258}{69}
%e A035470                                  {1458}{2367}      {1356}{249}{78}
%e A035470                                  {1467}{2358}      {159}{2346}{78}
%e A035470                                  {1236}{48}{57}    {159}{267}{348}
%e A035470                                  {138}{246}{57}    {168}{249}{357}
%e A035470                                  {156}{237}{48}    {18}{27}{36}{45}{9}
%e A035470                                  {18}{27}{36}{45}
%e A035470 (End)
%p A035470 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
%t A035470 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]];
%t A035470 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 &]}]];
%t A035470 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 25}] (* _Jean-François Alcover_, Mar 22 2017, after _Alois P. Heinz_ *)
%t A035470 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A035470 Table[Length[Select[sps[Range[n]],SameQ@@Total/@#&]],{n,0,10}] (* _Gus Wiseman_, Jul 13 2019 *)
%Y A035470 Cf. A164977, A164978. - _Alois P. Heinz_, Sep 03 2009
%Y A035470 Row sums of A275714.
%Y A035470 Row sums of A320438.
%Y A035470 Cf. A000110, A007837, A038041, A112956, A275780, A275781, A321455, A326512, A326513, A326518, A326534.
%K A035470 nonn
%O A035470 1,3
%A A035470 _Erich Friedman_
%E A035470 More terms from _John W. Layman_, Mar 18 2002
%E A035470 a(19)-a(33) from _Alois P. Heinz_, Sep 03 2009
%E A035470 a(34) from _Alois P. Heinz_, May 24 2015
%E A035470 a(35)-a(38) from _Max Alekseyev_, Feb 15 2024