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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
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
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 06 2014
STATUS
approved