

A236832


Number of ways to write 2*n  1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.


5



0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 3, 2, 4, 3, 3, 4, 3, 4, 4, 4, 4, 3, 5, 5, 7, 6, 3, 5, 4, 5, 4, 5, 6, 6, 6, 3, 5, 7, 6, 6, 3, 5, 8, 8, 8, 6, 7, 8, 7, 6, 5, 8, 9, 10, 5, 7, 9, 10, 11, 5, 8, 9, 9, 11, 6, 8, 9, 10, 8, 2, 9, 10, 9, 11, 6, 8, 11, 12, 7, 7, 10, 9, 10, 8, 7, 11, 10, 11, 6, 8, 12, 14, 13, 8, 10, 11, 12, 12, 10
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OFFSET

1,5


COMMENTS

Conjecture: a(n) > 0 for all n > 3.
This is stronger than Goldbach's weak conjecture which was finally proved by H. A. Helfgott in 2013.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252, 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897, 2013.
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(4) = 1 since 2*4  1 = 2 + 2 + 3 with 2 and 3 terms of A234695.
a(5) = 2 since 2*5  1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.


MATHEMATICA

p[n_]:=PrimeQ[Prime[n]n+1]
q[n_]:=PrimeQ[n]&&p[n]
a[n_]:=Sum[If[p[Prime[i]]&&p[Prime[j]]&&q[2n1Prime[i]Prime[j]], 1, 0], {i, 1, PrimePi[(2n1)/3]}, {j, i, PrimePi[(2n1Prime[i])/2]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A068307, A230219, A234695, A235189.
Sequence in context: A240975 A242166 A068211 * A089050 A167439 A272314
Adjacent sequences: A236829 A236830 A236831 * A236833 A236834 A236835


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 31 2014


STATUS

approved



