A239330 Number of odd primes p <= n with pi(n*(p+1)/2) - pi(n*(p-1)/2) prime, where pi(x) denotes the number of primes not exceeding x.

0, 0, 0, 1, 1, 2, 2, 3, 3, 2, 4, 2, 5, 3, 3, 4, 1, 3, 4, 4, 5, 4, 4, 4, 4, 3, 3, 5, 5, 5, 3, 6, 8, 5, 5, 3, 5, 6, 4, 4, 7, 6, 4, 4, 3, 5, 3, 4, 3, 5, 4, 4, 3, 3, 4, 2, 4, 2, 4, 4, 3, 4, 9, 3, 7, 4, 6, 4, 5, 5, 7, 4, 9, 9, 7, 7, 11, 7, 8, 8

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 5, 17.

We have verified this for n up to 3*10^5.

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(4) = 1 since 3 and pi(4*(3+1)/2) - pi(4*(3-1)/2) = pi(8) - pi(4) = 4 - 2 = 2 are both prime.
a(5) = 1 since 5 and pi(5*(5+1)/2) - pi(5*(5-1)/2) = pi(15) - pi(10) = 6 - 4 = 2 are both prime.
a(17) = 1 since 11 and pi(17*(11+1)/2) - pi(17*(11-1)/2) = pi(102) - pi(85) = 26 - 23 = 3 are both prime.

Mathematica

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

Crossrefs

Cf. A000040, A000720, A237578, A238278, A239328.

Sequence in context: A115074 A234399 A214749 * A276273 A039643 A288887

Adjacent sequences: A239327 A239328 A239329 * A239331 A239332 A239333

Keyword

nonn

Author

Zhi-Wei Sun, Mar 16 2014

Status

approved