OFFSET
1,10
COMMENTS
It is not true that a(2k+1) is always 0.
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
FORMULA
EXAMPLE
One of the three solutions for n = 10: 3 + 17 + 23 = 2 + 5 + 7 + 29 = 11 + 13 + 19.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, 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
STATUS
approved