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%I #13 Feb 19 2015 00:10:02
%S 0,0,0,0,1,0,1,1,2,2,1,1,2,2,2,1,2,0,1,2,3,3,1,2,3,3,3,2,2,1,1,4,2,2,
%T 3,2,4,3,3,3,2,3,4,3,4,2,4,2,2,3,2,5,3,2,4,5,5,5,4,4,1,4,5,2,4,2,4,3,
%U 3,4,4,2,5,3,5,1,5,3,0,6,4,5,4,2,6,4,5
%N Number of decompositions of 2n into sums of two primes p1 < p2 such that p2-p1-1 is also a prime.
%C a(n)=0 for n = 1, 2, 3, 4, 6, 18, 79. It is conjectured that there is not any other n for which a(n)=0.
%H Lei Zhou, <a href="/A254687/b254687.txt">Table of n, a(n) for n = 1..10000</a>
%H Lei Zhou, <a href="/A254687/a254687.jpg">Plot of a(n)</a> for n<=20000.
%e n=5, 2n=10=3+7. 7-3-1=3 is prime, so a(5)=1;
%e n=6, 2n=12=5+7. 7-5-1=1 is not prime, so a(6)=0;
%e ...
%e n=21, 2n=42=5+37=11+31=13+29=19+23. 37-5-1=31 is prime, 31-11-1=19 is prime, 29-13-1=15 is composite, 23-19-1=3 is prime: three primes in the form of p2-p1-1 found, so a(21)=3.
%t Table[e = 2 n; ct = 0; p1 = 1; While[p1 = NextPrime[p1]; p1 < n, p2 = e - p1; If[PrimeQ[p2], If[PrimeQ[Abs[p2 - p1 - 1]], ct++]]]; ct, {n, 1, 100}]
%o (Python)
%o from sympy import isprime, nextprime
%o def A254687(n):
%o ....y, x, n2 = 0, 2, 2*n
%o ....while x < n:
%o ........if isprime(n2-x) and isprime(n2-2*x-1):
%o ............y += 1
%o ........x = nextprime(x)
%o ....return y # _Chai Wah Wu_, Feb 18 2015
%Y Cf. A045917, A254688.
%K nonn,easy
%O 1,9
%A _Lei Zhou_, Feb 05 2015
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