

A237284


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


10



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



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

ZhiWei 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], 20142016.


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[2nPrime[Prime[k]]], 1, 0], {k, 1, PrimePi[PrimePi[2n1]]}]
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

ZhiWei Sun, Feb 06 2014


STATUS

approved



