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A232502
Number of ways to write n = k + m (0 < k < m) with 2*prime(m) - prime(k) prime.
5
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
1,8
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.
LINKS
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.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 24 2013
STATUS
approved