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A238568 a(n) = |{0 < k < n: n^2 - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A022307 A029413 A237523 * A238421 A105154 A076447

Adjacent sequences:  A238565 A238566 A238567 * A238569 A238570 A238571

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 28 2014

STATUS

approved

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Last modified February 27 03:30 EST 2020. Contains 332299 sequences. (Running on oeis4.)