A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

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

Offset

1,2

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.

(ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.

The conjecture implies that there are infinitely many primes p with p - pi(pi(p)) (or p + pi(pi(p)) prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

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

Example

a(5) = 1 since prime(3*5) - pi(3*5) = 47 - 6 = 41 is prime.
a(28) = 1 since prime(18*28) - pi(18*28) = prime(504) - pi(504) = 3607 - 96 = 3511 is prime.

Mathematica

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

Crossrefs

Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.

Sequence in context: A320107 A190321 A338409 * A266968 A237593 A338169

Adjacent sequences: A238887 A238888 A238889 * A238891 A238892 A238893

Keyword

nonn

Author

Zhi-Wei Sun, Mar 06 2014

Status

approved