A254688 Number of decompositions of 2n into sums of two primes p1 < p2 such that p2-p1+1 is also a prime.

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, 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, 1, 4, 3, 6, 4, 3, 3, 3, 5, 3, 2, 5, 5, 4, 2, 4, 3, 4, 5

Offset

1,8

Comments

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.

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Lei Zhou, Plot of a(n) for n <= 20000.

Example

n=4, 2n=8=3+5. 5-3+1=3 is prime, so a(4)=1;
n=7, 2n=14=3+11. 11-3+1=9 is not prime, so a(7)=0;
...
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.

Mathematica

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}]

Prog

(Python)
from sympy import isprime, nextprime
def A254688(n):
....y, x, n2 = 0, 2, 2*n
....while x < n:
........if isprime(n2-x) and isprime(n2-2*x+1):
............y += 1
........x = nextprime(x)
....return y # Chai Wah Wu, Feb 18 2015

Crossrefs

Cf. A045917, A254687.

Sequence in context: A245936 A199596 A074265 * A002636 A353656 A196062

Adjacent sequences: A254685 A254686 A254687 * A254689 A254690 A254691

Keyword

nonn,easy

Author

Lei Zhou, Feb 05 2015

Status

approved