A185636 a(n) = |{0 <= k < n: n+k and n+k^2 are both prime}|.

0, 2, 2, 2, 1, 2, 3, 1, 2, 3, 2, 3, 3, 1, 4, 2, 2, 3, 2, 2, 2, 5, 3, 3, 5, 2, 5, 6, 3, 4, 3, 3, 4, 4, 3, 5, 9, 3, 6, 5, 2, 6, 5, 3, 5, 3, 4, 5, 3, 4, 3, 4, 4, 4, 7, 3, 9, 14, 2, 8, 2, 4, 8, 6, 4, 3, 8, 2, 5, 9, 4, 7, 5, 2, 6, 4, 6, 12, 6, 4, 4, 7, 4, 8, 8, 3, 6, 8, 4, 8, 8, 5, 11, 4, 6, 5, 11, 7, 12, 10

Offset: 1

Author

Zhi-Wei Sun, Dec 18 2012

Keywords

• nonn

Comments

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

This conjecture has been verified for n up to 10^8. It is stronger than Bertrand's postulate proved by Chebyshev in 1850.

Zhi-Wei Sun also guessed the following refinement of the conjecture: For any integer n > 1456 there is an integer k among 0,...,n-1 such that n-k, n+k and n+k^2 are all prime; in other words, there is a prime p <= n such that 2n-p and n+(n-p)^2 are both prime.

For other refinements of the conjecture, the reader may consult arXiv:1211.1588.

The author also conjectured the following polynomial analogs:

(i) For a given integer polynomial f(x) of degree n > 0, there is an integer polynomial g(x) of degree at most n such that f(x)+g(x) and f(x)+g(x)^2 are both irreducible in Z[x].

(ii) Let F be a field with characteristic different from 2 and 3. If f(x) is an irreducible polynomial over F with degree n > 0, then there is a polynomial g(x) over F with deg(g) <= n such that f(x)+g(x)^2 is irreducible over F.

In a 2017 paper, the author announced a USD $100 prize for the first solution to his conjecture that for each n = 1,2,3,... there is an integer k among 0,...,n such that n+k and n+k^2 are both prime. - Zhi-Wei Sun, Dec 03 2017

Examples

			a(14)=1 since 3 is the only k among 0,...,13 with 14+k and 14+k^2 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2015.
• Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT].)

Crossrefs

Cf. A035250.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[n+k]==True&&PrimeQ[n+k^2]==True,1,0],{k,0,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]
nk[n_]:=Count[Range[0,n-1],?(And@@PrimeQ[n+{#,#^2}]&)]; Array[nk,100] (* _Harvey P. Dale, Jun 17 2014 *)