A163850 Primes p such that their distance to the nearest cube above p and also their distance to the nearest cube below p are prime.

3, 127, 24391, 29789, 328511, 2460373, 3048623, 9393929, 10503461

Offset

1,1

Comments

The two sequences A048763(p) and A048762(p), p=A000040(n), define

nearest cubes above and below each prime p. If p is in A146318, the

distance to the larger cube, A048763(p)-p, is prime. If p is

in the set {3, 11, 13, 19, 29, 67,...,107, 127, 223,..}, the distance to the lower

cube is prime. If both of these distances are prime, we insert p into the sequence.

Links

Table of n, a(n) for n=1..9.

Example

p=3 is in the sequence because the distance p-1=2 to the cube 1^3 below 3, and also the distance 8-p=5 to the cube 8=2^3 above p are prime.
p=127 is in the sequence because the distance p-125=2 to the cube 125=5^3 below p, and also the distance 216-p=89 to the cube 216=6^3 above p, are prime.

Mathematica

Clear[f, lst, p, n]; f[n_]:=IntegerPart[n^(1/3)]; lst={}; Do[p=Prime[n]; If[PrimeQ[p-f[p]^3]&&PrimeQ[(f[p]+1)^3-p], AppendTo[lst, p]], {n, 9!}]; lst
dncQ[n_]:=Module[{c=Floor[Surd[n, 3]]}, AllTrue[{n-c^3, (c+1)^3-n}, PrimeQ]]; Select[Prime[Range[230000]], dncQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 16 2016 *)

Crossrefs

Cf. A163848

Sequence in context: A071151 A041867 A134711 * A258671 A243213 A264582

Adjacent sequences: A163847 A163848 A163849 * A163851 A163852 A163853

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Extensions

Edited, first 5 entries checked by R. J. Mathar, Aug 12 2009

Two more terms (a(8) and a(9)) from Harvey P. Dale, Oct 16 2016

Status

approved