A327449 Number of ways the first n primes can be partitioned into three sets with equal sums.

0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 423, 0, 2624, 0, 13474, 0, 0, 0, 611736, 0, 4169165, 0, 30926812, 0, 214975174, 0, 1590432628, 0, 11431365932, 0, 83946004461, 0, 0, 0, 4615654888831, 0, 35144700468737, 0, 271133285220726, 0, 2103716957561013, 0, 0, 0, 0, 0, 990170108748552983, 0, 7855344215856348141

Offset: 1

Author

N. J. A. Sloane, Sep 19 2019

Keywords

• nonn

Comments

It is not true that a(2k+1) is always 0.

Examples

			One of the three solutions for n = 10: 3 + 17 + 23 = 2 + 5 + 7 + 29 = 11 + 13 + 19.
		

References

• Keith F. Lynch, Posting to Math Fun Mailing List, Sep 17 2019.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..102

Crossrefs

Cf. A000040, A007504, A022894, A112972, A275714, A103208, A111320, A113263, A327448, A327450.

Programs

Maple

s:= proc(n) option remember; `if`(n<2, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, x, y) option remember; `if`(n=1, 1, (p-> (l->
      add(`if`(p>l[i], 0, b(n-1, sort(subsop(i=l[i]-p, l))
      [1..2][])), i=1..3))([x, y, s(n)-x-y]))(ithprime(n)))
    end:
a:= n-> `if`(irem(2+s(n), 3, 'q')=0, b(n, q-2, q)/2, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 19 2019

Mathematica

s[n_] := s[n] = If[n < 2, 0, Prime[n] + s[n - 1]];
b[n_, x_, y_] := b[n, x, y] = If[n == 1, 1, Function[p, Function[l, Sum[If[ p > l[[i]], 0, b[n - 1, Sequence @@ Sort[ReplacePart[l, i -> l[[i]] - p]][[1;; 2]]]], {i, 1, 3}]][{x, y, s[n] - x - y}]][Prime[n]]];
a[n_] := If[Mod[2+s[n], 3]==0, q = Quotient[2+s[n], 3]; b[n, q-2, q]/2, 0];
Array[a, 40] (* Jean-François Alcover, Apr 09 2020, after Alois P. Heinz *)

PARI

EqSumThreeParts(v)={ my(n=#v, vs=vector(n), m=vecsum(v)/3, brk=0);
  for(i=1, n-1, vs[i+1]=vs[i]+v[i]; if(vs[i]<=min(1000,m), brk=i));
  my(q=Vecrev(prod(i=1, brk, 1+x^v[i]+y^v[i])));
  my(recurse(k,s,p)=if(k==brk, if(s<#q, polcoef(p*q[s+1],m,y)), if(s<=vs[k], self()(k-1, s, p*(1 + y^v[k]))) + if(s>=v[k], self()(k-1, s-v[k], p)) ));
  if(frac(m), 0, recurse(n-1, m, 1 + O(y*y^m))/2);
}
a(n)={EqSumThreeParts(primes(n))} \\ Andrew Howroyd, Sep 19 2019

Formula

a(n) > 0 <=> n in { A103208 }, with odd n in { A111320 }. - Alois P. Heinz, Sep 19 2019

Extensions

Corrected and a(30)-a(52) added by Andrew Howroyd, Sep 19 2019

a(53) and beyond from Alois P. Heinz, Sep 19 2019