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A035470 Number of ways to break {1,2,3,...n} into sets with equal sums. 32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n) = 2 <=> |{d|n*(n+1)/2 : d>=n}| = 2. - Alois P. Heinz, Sep 03 2009
LINKS
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.
Sequence in context: A078014 A063867 A024723 * A061292 A138068 A246052
KEYWORD
nonn
AUTHOR
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

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)