A237284 Number of ordered ways to write 2*n = p + q with p, q and A000720(p) all prime.

0, 0, 1, 2, 2, 1, 2, 3, 2, 2, 4, 3, 1, 3, 2, 1, 5, 3, 1, 3, 3, 3, 4, 5, 2, 3, 4, 1, 4, 3, 3, 6, 2, 1, 6, 6, 3, 4, 7, 1, 4, 6, 3, 5, 6, 2, 4, 4, 2, 6, 5, 3, 5, 4, 3, 7, 8, 2, 4, 8, 1, 4, 5, 3, 6, 5, 4, 2, 7, 5, 6, 6, 3, 4, 6, 2, 5, 7, 2, 4

Offset

1,4

Comments

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 6, 13, 16, 19, 28, 34, 40, 61, 166, 278.

This is stronger than Goldbach's conjecture.

The conjecture is true for n <= 5*10^8. - Dmitry Kamenetsky, Mar 13 2020

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Carlos Rivera, Conjecture 85. Conjectures stricter that the Goldbach ones, Prime Puzzles

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Example

a(13) = 1 since 2*13 = 3 + 23 with 3, 23 and A000720(3) = 2 all prime.
a(278) = 1 since 2*278 = 509 + 47 with 509, 47 and A000720(509) = 97 all prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[2n-Prime[Prime[k]]], 1, 0], {k, 1, PrimePi[PrimePi[2n-1]]}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000720, A002372, A002375, A006450, A236566.

Sequence in context: A025833 A200647 A261625 * A294186 A294185 A035462

Adjacent sequences: A237281 A237282 A237283 * A237285 A237286 A237287

Keyword

nonn

Author

Zhi-Wei Sun, Feb 06 2014

Status

approved