

A185636


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


15



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
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OFFSET

1,2


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.
ZhiWei Sun also guessed the following refinement of the conjecture: For any integer n > 1456 there is an integer k among 0,...,n1 such that nk, n+k and n+k^2 are all prime; in other words there is a prime p <= n such that 2np and n+(np)^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.  ZhiWei Sun, Dec 03 2017


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..20000
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 20142015.
Z.W. 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. 279310. (See also arXiv:1211.1588 [math.NT].)


EXAMPLE

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


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[n+k]==True&&PrimeQ[n+k^2]==True, 1, 0], {k, 0, n1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
nk[n_]:=Count[Range[0, n1], _?(And@@PrimeQ[n+{#, #^2}]&)]; Array[nk, 100] (* Harvey P. Dale, Jun 17 2014 *)


CROSSREFS

Cf. A035250.
Sequence in context: A264051 A120965 A151931 * A182597 A290491 A194314
Adjacent sequences: A185633 A185634 A185635 * A185637 A185638 A185639


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 18 2012


STATUS

approved



