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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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei 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[2n-1-Prime[i]-Prime[j]], 1, 0], {i, 1, PrimePi[(2n-1)/3]}, {j, i, PrimePi[(2n-1-Prime[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

Zhi-Wei Sun, Jan 31 2014

STATUS

approved

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Last modified May 7 14:09 EDT 2021. Contains 343650 sequences. (Running on oeis4.)