A307877 -id:A307877 - cp's OEIS frontend

A306443 Number of ways of partitioning the set of the first n primes into two subsets whose sums differ at most by 1.

1, 0, 1, 1, 1, 1, 3, 2, 6, 5, 16, 13, 45, 39, 138, 122, 439, 392, 1417, 1286, 4698, 4341, 16021, 14860, 55146, 51085, 190274, 178402, 671224, 634511, 2404289, 2260918, 8535117, 8067237, 30635869, 29031202, 110496946, 105250449, 401422210, 383579285, 1467402238

Offset: 0

Alois P. Heinz, May 31 2019

nonn

			a(8) = 6: 2,17,19/3,5,7,11,13; 3,5,11,19/2,7,13,17; 3,5,13,17/2,7,11,19; 3,7,11,17/2,5,13,19; 2,3,5,11,17/7,13,19; 2,5,7,11,13/3,17,19.
a(9) = 5: 2,3,5,17,23/7,11,13,19; 2,5,7,13,23/3,11,17,19; 2,5,7,17,19/3,11,13,23; 2,5,11,13,19/3,7,17,23; 2,7,11,13,17/3,5,19,23.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Wikipedia, Partition problem

Bisections give: A022894 (odd part), A113040 (even part).

Cf. A000040, A069918, A307877.

s:= proc(n) s(n):= `if`(n=0, 1, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
     `if`(n>s(i), 0, (p->b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(i))))
    end:
a:= n-> ceil(b(0, n)/2):
seq(a(n), n=0..45);

s[n_] := s[n] = If[n == 0, 1, Prime[n] + s[n - 1]];
b[n_, i_] := b[n, i] = If[i==0, If[n <= 1, 1, 0], If[n > s[i], 0, Function[ p, b[n + p, i - 1] + b[Abs[n - p], i - 1]][Prime[i]]]];
a[n_] := Ceiling[b[0, n]/2];
a /@ Range[0, 45] (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

A308682 Number of ways of partitioning the set of the first n positive triangular numbers into two subsets whose sums differ at most by 1.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 2, 7, 6, 8, 13, 42, 33, 52, 105, 318, 310, 485, 874, 3281, 2974, 5240, 9488, 34233, 30418, 55715, 104730, 378529, 352467, 642418, 1193879, 4466874, 4165910, 7762907, 14493951, 54162165, 50621491, 95133799, 179484713, 674845081

Offset: 0

Alois P. Heinz, Jun 16 2019

nonn

			a(4) = 1: 1,3,6/10.
a(5) = 1: 1,6,10/3,15.
a(6) = 1: 1,6,21/3,10,15.
a(7) = 1: 1,3,10,28/6,15,21.
a(8) = 2: 1,6,10,15,28/3,21,36; 1,10,21,28/3,6,15,36.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Partition problem

Cf. A000217, A000292, A050407, A307877.

s:= proc(n) s(n):= `if`(n=0, 1, n*(n+1)/2+s(n-1)) end:
b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
     `if`(n>s(i), 0, (p->b(n+p, i-1)+b(abs(n-p), i-1))(i*(i+1)/2)))
    end:
a:= n-> ceil(b(0, n)/2):
seq(a(n), n=0..45);

s[n_] := s[n] = If[n == 0, 1, n(n+1)/2 + s[n-1]];
b[n_, i_] := b[n, i] = If[i == 0, If[n <= 1, 1, 0], If[n > s[i], 0, Function[p, b[n + p, i-1] + b[Abs[n-p], i-1]][i(i+1)/2]]];
a[n_] := Ceiling[b[0, n]/2];
a /@ Range[0, 45] (* Jean-François Alcover, May 04 2020, translated from Maple *)

cp's OEIS Frontend

A306443 Number of ways of partitioning the set of the first n primes into two subsets whose sums differ at most by 1.

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