A237595 a(n) = |{1 <= k <= n: n + pi(k^2) is prime}|, where pi(.) is given by A000720.

0, 1, 3, 0, 3, 1, 3, 3, 3, 1, 5, 2, 6, 3, 4, 2, 6, 3, 7, 3, 2, 6, 8, 1, 10, 3, 5, 8, 9, 2, 9, 6, 3, 5, 14, 5, 11, 6, 9, 3, 13, 8, 11, 8, 8, 6, 8, 8, 11, 9, 6, 12, 15, 10, 11, 5, 11, 12, 13, 9, 12, 9, 5, 17, 15, 9, 18, 13, 11, 12

Offset

1,3

Comments

Conjecture: (i) a(n) > 0 for all n > 4.

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

(iii) For any integer n > 1, there is a positive integer k <= (n+1)/2 such that pi(n + k*(k+1)/2) is prime.

(iv) Any integer n > 1 can be written as p + pi(k*(k+1)/2), where p is a prime and k is among 1, ..., n-1.

Links

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(2) = 1 since 2 + pi(1^2) = 2 is prime.
a(6) = 1 since 6 + pi(6^2) = 6 + 11 = 17 is prime.
a(10) = 1 since 10 + pi(5^2) = 10 + 9 = 19 is prime.
a(21) = 2 since 21 + pi(2^2) = 23 and 21 + pi(9^2) = 43 are both prime.
a(24) = 1 since 24 + pi(21^2) = 24 + 85 = 109 is prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[n+PrimePi[k^2]], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 70}]

Crossrefs

Cf. A000040, A000720, A237453, A237496, A237497, A237578, A237582.

Sequence in context: A318504 A343877 A318505 * A322575 A166445 A298082

Adjacent sequences: A237592 A237593 A237594 * A237596 A237597 A237598

Keyword

nonn

Author

Zhi-Wei Sun, Feb 09 2014

Status

approved