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A035470
Number of ways to break {1,2,3,...,n} into sets with equal sums.
66
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
1,3
COMMENTS
a(n) = 2 <=> |{d|n*(n+1)/2 : d>=n}| = 2. - Alois P. Heinz, Sep 03 2009
EXAMPLE
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)
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 *)
CROSSREFS
Cf. A164977, A164978. - Alois P. Heinz, Sep 03 2009
Row sums of A275714.
Row sums of A320438.
Sequence in context: A078014 A063867 A024723 * A061292 A138068 A246052
KEYWORD
nonn,changed
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
STATUS
approved