A239328 Number of primes p < n with pi(p*n) - pi((p-1)n) prime, where pi(x) denotes the number of primes not exceeding x.

0, 0, 0, 1, 1, 2, 3, 2, 4, 3, 4, 4, 5, 5, 3, 4, 1, 2, 2, 5, 2, 2, 2, 5, 3, 3, 5, 8, 4, 7, 5, 5, 6, 5, 4, 4, 4, 6, 7, 5, 4, 6, 5, 8, 7, 4, 1, 2, 3, 8, 5, 3, 8, 5, 7, 6, 6, 4, 6, 9, 3, 6, 7, 8, 6, 3, 4, 4, 5, 6, 7, 6, 11, 10, 4, 6, 6, 9, 6, 5

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n > 3. Also, for any integer n > 4, there is a prime p < n with pi((p+1)*n) - pi(p*n) prime.

We have verified that a(n) > 0 for all n = 4, ..., 7*10^5.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

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

Example

a(4) = 1 since 2 and pi(2*4) - pi(1*4) = 4 - 2 = 2 are both prime.
a(5) = 1 since 3 and pi(3*5) - pi(2*5) = 6 - 4 = 2 are both prime.
a(17) = 1 since 11 and pi(11*17) - pi(10*17) = 42 - 39 = 3 are both prime.
a(47) = 1 since 37 ad pi(37*47) - pi(36*47) = 270 - 263 = 7 are both prime.

Mathematica

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

Crossrefs

Cf. A000040, A000720, A237578, A238277, A238278, A238281, A239330.

Sequence in context: A173753 A295753 A144968 * A076753 A352576 A245134

Adjacent sequences: A239325 A239326 A239327 * A239329 A239330 A239331

Keyword

nonn

Author

Zhi-Wei Sun, Mar 16 2014

Status

approved