cp's OEIS Frontend

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

%I A232502 #15 Jul 11 2018 05:31:51
%S A232502 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,
%T A232502 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,
%U A232502 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
%N A232502 Number of ways to write n = k + m (0 < k < m) with 2*prime(m) - prime(k) prime.
%C A232502 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.
%C A232502 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.
%C A232502 (ii) Any integer n > 4, can be written as k + m (0 < k < m) with 2*prime(m) + prime(k) prime.
%H A232502 Zhi-Wei Sun, <a href="/A232502/b232502.txt">Table of n, a(n) for n = 1..10000</a>
%H A232502 B. Green and T. Tao, <a href="https://dx.doi.org/10.4007/annals.2008.167.481">The primes contain arbitrarily long arithmetic progressions</a>, Annals of Math. 167(2008), 481-547.
%H A232502 J. G. van der Corput, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002279932">Über Summen von Primzahlen und Primzahlquadraten</a>, Math. Ann. 116 (1939), 1-50.
%e A232502 a(17) = 1 since 2*prime(10) - prime(7) = 2*29 - 17 = 41 is prime.
%e A232502 a(21) = 1 since 2*prime(19) - prime(2) = 2*67 - 3 = 131 is prime.
%e A232502 a(34) = 1 since 2*prime(24) - prime(10) = 2*89 - 29 = 149 is prime.
%e A232502 a(43) = 1 since 2*prime(28) - prime(15) = 2*107 - 47 = 167 is prime.
%t A232502 a[n_]:=Sum[If[PrimeQ[2*Prime[n-k]-Prime[k]],1,0],{k,1,(n-1)/2}]
%t A232502 Table[a[n],{n,1,100}]
%Y A232502 Cf. A000040, A232463, A232465.
%K A232502 nonn
%O A232502 1,8
%A A232502 _Zhi-Wei Sun_, Nov 24 2013