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

1,12

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A023200, A046132, A218829.

Sequence in context: A172069 A054348 A239660 * A373360 A037813 A159700

Adjacent sequences: A237345 A237346 A237347 * A237349 A237350 A237351

Keyword

nonn

Author

Zhi-Wei Sun, Feb 06 2014

Status

approved