A238568 a(n) = |{0 < k < n: n^2 - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

0, 1, 1, 1, 2, 2, 2, 1, 2, 1, 3, 2, 4, 3, 4, 2, 2, 5, 5, 3, 4, 4, 8, 1, 3, 3, 4, 3, 4, 3, 6, 3, 4, 4, 3, 4, 6, 3, 5, 2, 1, 8, 3, 10, 6, 5, 5, 9, 7, 6, 3, 8, 7, 9, 2, 5, 5, 2, 2, 9, 7, 3, 5, 8, 7, 6, 8, 7, 9, 9, 6, 3, 7, 8, 14, 5, 9, 10, 8, 11

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 8, 10, 24, 41.

(ii) For any integer n > 6, there is a positive integer k < n with n^2 + pi(k*n) - 1 prime.

(iii) If n > 2, then pi(n^2) - pi(k*n) is prime for some 0 < k < n. If n > 1, then pi(n^2) + pi(k*n) - 1 is prime for some 0 < k < n.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Example

a(2) = 1 since 2^2 - pi(1*2) = 4 - 1 = 3 is prime.
a(3) = 1 since 3^2 - pi(1*3) = 9 - 2 = 7 is prime.
a(4) = 1 since 4^2 - pi(3*4) = 16 - 5 = 11 is prime.
a(8) = 1 since 8^2 - pi(4*8) = 64 - 11 = 53 is prime.
a(10) = 1 since 10^2 - pi(6*10) = 100 - 17 = 83 is prime.
a(24) = 1 since 24^2 - pi(14*24) = 576 - 67 = 509 is prime.
a(41) = 1 since 41^2 - pi(10*41) = 1681 - 80 = 1601 is prime.

Mathematica

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

Crossrefs

Cf. A000040, A000720, A237578, A237615, A237712, A238570.

Sequence in context: A029413 A237523 A339812 * A368148 A238421 A105154

Adjacent sequences: A238565 A238566 A238567 * A238569 A238570 A238571

Keyword

nonn

Author

Zhi-Wei Sun, Feb 28 2014

Status

approved