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

%I A306443 #65 Apr 30 2020 07:51:08
%S A306443 1,0,1,1,1,1,3,2,6,5,16,13,45,39,138,122,439,392,1417,1286,4698,4341,
%T A306443 16021,14860,55146,51085,190274,178402,671224,634511,2404289,2260918,
%U A306443 8535117,8067237,30635869,29031202,110496946,105250449,401422210,383579285,1467402238
%N A306443 Number of ways of partitioning the set of the first n primes into two subsets whose sums differ at most by 1.
%H A306443 Alois P. Heinz, <a href="/A306443/b306443.txt">Table of n, a(n) for n = 0..2000</a>
%H A306443 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_problem">Partition problem</a>
%e A306443 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.
%e A306443 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.
%p A306443 s:= proc(n) s(n):= `if`(n=0, 1, ithprime(n)+s(n-1)) end:
%p A306443 b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
%p A306443      `if`(n>s(i), 0, (p->b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(i))))
%p A306443     end:
%p A306443 a:= n-> ceil(b(0, n)/2):
%p A306443 seq(a(n), n=0..45);
%t A306443 s[n_] := s[n] = If[n == 0, 1, Prime[n] + s[n - 1]];
%t A306443 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]]]];
%t A306443 a[n_] := Ceiling[b[0, n]/2];
%t A306443 a /@ Range[0, 45] (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y A306443 Bisections give: A022894 (odd part), A113040 (even part).
%Y A306443 Cf. A000040, A069918, A307877.
%K A306443 nonn
%O A306443 0,7
%A A306443 _Alois P. Heinz_, May 31 2019