A232502 Number of ways to write n = k + m (0 < k < m) with 2*prime(m) - prime(k) prime.

0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 2, 3, 1, 5, 2, 3, 1, 4, 5, 3, 4, 2, 3, 2, 3, 5, 5, 3, 7, 1, 5, 4, 8, 3, 4, 5, 6, 5, 1, 6, 4, 9, 3, 8, 4, 6, 3, 10, 5, 8, 4, 8, 3, 9, 6, 4, 4, 4, 10, 6, 10, 4, 11, 5, 11, 6, 9, 5, 10, 9, 8, 6, 9, 7, 9, 11, 9, 11, 5, 10, 9, 12, 6, 6, 10, 9, 8, 13, 4, 12, 10, 12, 8, 7, 12, 14

Offset: 1

Author

Zhi-Wei Sun, Nov 24 2013

Keywords

• nonn

Comments

Note that prime(k), prime(m), 2*prime(m) - prime(k) form a three-term arithmetic progression. It is known that there are infinitely many nontrivial three-term arithmetic progressions whose terms are all prime.

Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 7, 9, 11, 14, 17, 21, 34, 43.

(ii) Any integer n > 4, can be written as k + m (0 < k < m) with 2*prime(m) + prime(k) prime.

Examples

			a(17) = 1 since 2*prime(10) - prime(7) = 2*29 - 17 = 41 is prime.
a(21) = 1 since 2*prime(19) - prime(2) = 2*67 - 3 = 131 is prime.
a(34) = 1 since 2*prime(24) - prime(10) = 2*89 - 29 = 149 is prime.
a(43) = 1 since 2*prime(28) - prime(15) = 2*107 - 47 = 167 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167(2008), 481-547.
• J. G. van der Corput, Über Summen von Primzahlen und Primzahlquadraten, Math. Ann. 116 (1939), 1-50.

Crossrefs

Cf. A000040, A232463, A232465.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[2*Prime[n-k]-Prime[k]],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,100}]