A254688 - OEIS

%I #14 Feb 19 2015 00:10:09

%S 0,0,0,1,1,1,0,2,1,1,2,2,1,2,2,1,2,3,0,2,2,2,3,1,2,3,2,2,1,3,0,4,3,1,

%T 2,2,2,5,3,3,2,4,2,3,4,2,2,4,2,5,2,3,5,1,2,5,4,4,3,5,2,4,4,2,4,3,3,4,

%U 1,4,3,6,4,3,3,3,5,3,2,5,5,4,2,4,3,4,5

%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, 7, 19, 31, 139. It is conjectured that there is not any other n for which a(n)=0.

%H Lei Zhou, <a href="/A254688/b254688.txt">Table of n, a(n) for n = 1..10000</a>

%H Lei Zhou, <a href="/A254688/a254688.jpg">Plot of a(n)</a> for n <= 20000.

%e n=4, 2n=8=3+5. 5-3+1=3 is prime, so a(4)=1;

%e n=7, 2n=14=3+11. 11-3+1=9 is not prime, so a(7)=0;

%e ...

%e n=18 2n=36=5+31=7+29=13+23=17+19. 31-5+1=27 is composite, 29-7+1=23 is prime, 23-13+1=11 is prime, 19-17+1=3 is prime: three primes in the form of p2-p1+1 found, so a(18)=3.

%t Table[e = 2 n; ct = 0; p1 = 1; While[p1 = NextPrime[p1]; p1 < n, p2 = e - p1; If[PrimeQ[p2], If[PrimeQ[p2 - p1 + 1], ct++]]]; ct, {n, 1, 100}]

%o (Python)

%o from sympy import isprime, nextprime

%o def A254688(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, A254687.

%K nonn,easy

%O 1,8

%A _Lei Zhou_, Feb 05 2015