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A327449 Number of ways the first n primes can be partitioned into three sets with equal sums. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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

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-Fran├žois 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

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

Sequence in context: A308254 A111417 A007271 * A320179 A035656 A325675

Adjacent sequences:  A327446 A327447 A327448 * A327450 A327451 A327452

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

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Last modified January 28 05:27 EST 2022. Contains 350654 sequences. (Running on oeis4.)