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A327449
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Number of ways the first n primes can be partitioned into three sets with equal sums.
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5
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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
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OFFSET
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1,10
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COMMENTS
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It is not true that a(2k+1) is always 0.
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REFERENCES
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Keith F. Lynch, Posting to Math Fun Mailing List, Sep 17 2019.
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LINKS
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FORMULA
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EXAMPLE
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One of the three solutions for n = 10: 3 + 17 + 23 = 2 + 5 + 7 + 29 = 11 + 13 + 19.
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MAPLE
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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):
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MATHEMATICA
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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];
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PROG
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(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);
}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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