A154669 Averages k of twin prime pairs such that 2*k^3 + 12*k^2 is a square.

12, 282, 642, 1452, 12162, 17292, 34842, 98562, 194682, 233922, 347772, 383682, 410412, 544962, 749082, 1071642, 1302492, 1421292, 1503372, 1685442, 2794242, 3011052, 3235962, 3312732, 3792252, 3875322, 4755522

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

Keywords

• nonn

Comments

(11,13) is a twin prime pair with average 12; sqrt(2*12^3 + 12*12^2) = 72.

Crossrefs

Cf. A152811.

Programs

Maple

a := proc (n) if isprime(n-1) = true and isprime(n+1) = true and type(sqrt(2*n^3+12*n^2), integer) = true then n else end if end proc: seq(a(n), n = 3 .. 5000000); # Emeric Deutsch, Jan 20 2009

Mathematica

a[n_]:=Sqrt[2*n^3+12*n^2];lst={};Do[If[Floor[a[n]]==a[n],If[PrimeQ[n-1]&&PrimeQ[n+1],AppendTo[lst,n]]],{n,9!}];lst
Select[Mean/@Select[Partition[Prime[Range[400000]],2,1],#[[2]]-#[[1]] == 2&],IntegerQ[Sqrt[2#^3+12#^2]]&] (* Harvey P. Dale, Sep 05 2017 *)

Extensions

Extended by Emeric Deutsch, Jan 20 2009