A224030 a(n) = |{0

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

Offset: 1

Author

Zhi-Wei Sun, Apr 15 2013

Keywords

• nonn

Comments

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

This has been verified for n up to 10^8.

We also conjecture that for any integer n>1 there is an integer 0

Examples

			a(7) = 1 since 2*7+5 = 19 and 2*7^3+5^3 = 811 are both prime.
a(57) = 1 since 2*57+23 = 137 and 2*57^3+23^3 = 382553 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A185636, A204065, A220413.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[2n+k]==True&&PrimeQ[2n^3+k^3]==True,1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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