A237348 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 4 and prime(prime(m)) + 4 are both prime.

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 2, 3, 3, 3, 5, 4, 2, 4, 1, 5, 1, 5, 1, 4, 4, 3, 3, 3, 1, 5, 4, 4, 3, 5, 3, 5, 6, 3, 3, 4, 3, 4, 5, 1, 5, 3, 3, 3, 5, 4, 2, 8, 1, 2, 5, 6

Offset: 1

Author

Zhi-Wei Sun, Feb 06 2014

Keywords

• nonn

Comments

Conjecture: For each d = 1, 2, 3, ... there is a positive integer N(d) for which any integer n > N(d) can be written as k + m with k > 0 and m > 0 such that prime(k) + 2*d and prime(prime(m)) + 2*d are both prime. In particular, we may take (N(1), N(2), ..., N(10)) = (2, 11, 4, 15, 31, 4, 2, 77, 4, 7).

This extension of the "Super Twin Prime Conjecture" (posed by the author) implies de Polignac's well-known conjecture that any positive even number can be a difference of two primes infinitely often.

Examples

			a(7) = 1 since 7 = 6 + 1 with prime(6) + 4 = 13 + 4 = 17 and prime(prime(1)) + 4 = prime(2) + 4 = 7 both prime.
a(114) = 1 since 114 = 78 + 36 with prime(78) + 4 = 397 + 4 = 401 and prime(prime(36)) + 4 = prime(151) + 4 = 877 + 4 = 881 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.

Crossrefs

Cf. A000040, A023200, A046132, A218829.

Programs

Mathematica

pq[n_]:=pq[n]=PrimeQ[Prime[n]+4]
PQ[n_]:=PrimeQ[Prime[Prime[n]]+4]
a[n_]:=Sum[If[pq[k]&&PQ[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]