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A237291 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720). 4
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 1, 2, 0, 2, 3, 1, 0, 2, 2, 1, 3, 2, 1, 1, 1, 1, 2, 3, 2, 2, 0, 3, 4, 2, 2, 3, 2, 1, 3, 4, 1, 5, 2, 1, 2, 3, 4, 3, 1, 1, 3, 2, 2, 4, 3, 2, 3, 3, 1, 5, 5, 1, 3, 4, 2, 3, 4, 4, 2, 4, 2, 3, 4, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

Conjecture: a(n) > 0 for all n > 36.

This is stronger than Goldbach's weak conjecture 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(16) = 1 since 2*16 - 1 = 3 + 11 + 17 with 3, 11, 17, pi(3) = 2, pi(11) = 5 and pi(17) = 7 all prime.

a(179) = 1 since 2*179 - 1 = 83 + 83 + 191 with 83, 191, pi(83) = 23 and pi(191) = 43 all prime.

MATHEMATICA

p[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]

a[n_]:=Sum[If[p[2n-1-Prime[Prime[i]]-Prime[Prime[j]]], 1, 0], {i, 1, PrimePi[PrimePi[(2n-1)/3]]}, {j, i, PrimePi[PrimePi[(2n-1-Prime[Prime[i]])/2]]}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A000720, A006450, A068307, A230219, A236832, A237284.

Sequence in context: A059018 A249371 A122190 * A146093 A091269 A287455

Adjacent sequences:  A237288 A237289 A237290 * A237292 A237293 A237294

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 06 2014

STATUS

approved

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Last modified May 22 22:13 EDT 2022. Contains 353959 sequences. (Running on oeis4.)