A224030 a(n) = |{0<k<n: 2*n+k and 2*n^3+k^3 are both prime}|.

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

1,5

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<k<n such that n^2+k^2 is 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.

Example

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.

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}]

Crossrefs

Cf. A185636, A204065, A220413.

Sequence in context: A159905 A283735 A274534 * A233136 A339717 A106054

Adjacent sequences: A224027 A224028 A224029 * A224031 A224032 A224033

Keyword

nonn

Author

Zhi-Wei Sun, Apr 15 2013

Status

approved