A236552 a(n) = |{0 < k < n: 6*k - 1, 6*k + 1, 6*k + 5 and prime(n-k) + 6 are all prime}|.

0, 0, 0, 1, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 4, 5, 4, 4, 3, 4, 3, 5, 5, 5, 2, 2, 4, 3, 4, 3, 3, 4, 5, 3, 5, 4, 6, 6, 6, 4, 4, 1, 3, 6, 5, 2, 4, 4, 5, 6, 8, 6, 5, 4, 8, 4, 4, 5, 7, 6, 4, 4, 4, 6, 6, 5, 5, 6, 6, 7, 6, 4, 3, 5, 7, 3, 4, 4, 3, 6, 5, 5, 6, 6, 7, 5, 4, 3, 5, 5, 3, 3, 2, 2

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 3.

(ii) For any integer n > 4, there is a positive integer k < n such that 6*k - 5, 6*k - 1, 6*k + 1 and prime(n-k) + 6 are all prime.

(iii) Any integer n > 7 can be written as p + q with q > 0 such that p, p + 6 and prime(q) + 6 are all prime.

(iv) Each integer n > 4 can be written as k*(k+1) + m with k > 0 and m > 0 such that prime(m) + 6 is prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(4) = 1 since 6*1 - 1, 6*1 + 1, 6*1 + 5 and prime(4-1) + 6 = 11 are all prime.
a(48) = 1 since 6*32 - 1 = 191, 6*32 + 1 = 193, 6*32 + 5 = 197 and prime(48-32) + 6 = 53 + 6 = 59 are all prime.

Mathematica

p[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[6n+5]
q[n_]:=PrimeQ[Prime[n]+6]
a[n_]:=Sum[If[p[k]&&q[n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A022004, A022005, A023201, A046117, A236462, A236464.

Sequence in context: A210681 A366544 A096520 * A352628 A080748 A305374

Adjacent sequences: A236549 A236550 A236551 * A236553 A236554 A236555

Keyword

nonn

Author

Zhi-Wei Sun, Jan 28 2014

Status

approved