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 A035470 Number of ways to break {1,2,3,...n} into sets with equal sums. 22
 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 (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. Cf. A000110, A007837, A038041, A112956, A275780, A275781, A321455, A326512, A326513, A326518, A326534. Sequence in context: A078014 A063867 A024723 * A061292 A138068 A246052 Adjacent sequences:  A035467 A035468 A035469 * A035471 A035472 A035473 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 STATUS approved

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Last modified October 16 11:28 EDT 2019. Contains 328056 sequences. (Running on oeis4.)