A238643 Number of primes p <= n such that 2*pi(p) - (-1)^n and p*n +((-1)^n - 3)/2 are both prime, where pi(x) is the number of primes not exceeding x.

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

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 2, then 2*p*n + 1 (or 2*p*n - 1) is prime for some prime p < n.

Part (i) of the conjecture is a further extension of the conjecture in A238597 to cover the even case.

Links

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

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Example

a(9) = 1 since 5, 2*pi(5)-(-1)^9 = 2*3 + 1 = 7 and 5*9 + ((-1)^9-3)/2 = 45 - 2 = 43 are all prime.
a(10) = 1 since 3, 2*pi(3)-(-1)^(10) = 2*2 - 1 = 3 and 3*10 + ((-1)^(10)-3)/2 = 30 - 1 = 29 are all prime.
a(268) = 1 since 23, 2*pi(23) - 1`= 2*9 - 1 = 17 and 23*268 - 1 = 6163 are all prime.
a(389) = 1 since 71, 2*pi(71) + 1 = 2*20 + 1 = 41 and 71*389 - 2 = 27617 are all prime.

Mathematica

p[n_, k_]:=PrimeQ[2k-(-1)^n]&&PrimeQ[n*Prime[k]+((-1)^n-3)/2]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[n]}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000720, A238580, A238597.

Sequence in context: A328582 A227153 A328581 * A140193 A073741 A071838

Adjacent sequences: A238640 A238641 A238642 * A238644 A238645 A238646

Keyword

nonn

Author

Zhi-Wei Sun, Mar 01 2014

Status

approved