A238570 a(n) = |{0 < k < n: pi((k+1)^2) - pi(k^2) and pi(n^2) - pi(k^2) are both prime}|, where pi(x) denotes the number of primes not exceeding x.

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

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 1.

(ii) If n > 4, then pi(n^2) + pi(k^2) is prime for some k = 2, ..., n-1.

(iii) If n > 0 is not a divisor of 12, then n^2 + pi(k^2) is prime for some k = 2, ..., n-1.

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(8) = 1 since pi(8^2) - pi(7^2) = 18 - 15 = 3 is prime.
a(61) = 1 since pi(27^2) - pi(26^2) = 129 - 122 = 7 and pi(61^2) - pi(26^2) = 519 - 122 = 397 are both prime.
a(86) = 1 since pi(3^2) - pi(2^2) = 4 - 2 = 2 and pi(86^2) - pi(2^2) = 939 - 2 = 937 are both prime.

Mathematica

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

Crossrefs

Cf. A000040, A000720, A237657, A237595, A238568.

Sequence in context: A096392 A332964 A105825 * A145425 A201909 A070352

Adjacent sequences: A238567 A238568 A238569 * A238571 A238572 A238573

Keyword

nonn

Author

Zhi-Wei Sun, Feb 28 2014

Status

approved