A237348 - OEIS

%I #12 Feb 06 2014 12:09:43

%S 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,

%T 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,

%U 5,3,3,3,5,4,2,8,1,2,5,6

%N 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.

%C 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).

%C 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.

%H Zhi-Wei Sun, <a href="/A237348/b237348.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b81b9aa9.1402">Super Twin Prime Conjecture</a>, a message to Number Theory List, Feb. 6, 2014.

%e 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.

%e 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.

%t pq[n_]:=pq[n]=PrimeQ[Prime[n]+4]

%t PQ[n_]:=PrimeQ[Prime[Prime[n]]+4]

%t a[n_]:=Sum[If[pq[k]&&PQ[n-k],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A023200, A046132, A218829.

%K nonn

%O 1,12

%A _Zhi-Wei Sun_, Feb 06 2014