A234240 Cubes which are arithmetic mean of two consecutive primes.

64, 1728, 4096, 17576, 21952, 46656, 110592, 195112, 287496, 314432, 405224, 474552, 1061208, 1191016, 1404928, 1601613, 1906624, 2000376, 2146689, 2197000, 3241792, 3511808, 4913000, 5268024, 6229504, 6751269, 6859000, 7077888, 11239424, 20346417, 21952000

Offset: 1

Author

K. D. Bajpai, Dec 21 2013

Keywords

• nonn

Examples

			64 is in the sequence because cube 64 = 4^3 = (61+67)/2 is arithmetic mean of two consecutive primes.
1728 is in the sequence because 1728 = 12^3 = (1723+1733)/2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (terms n = 1..57 from K. D. Bajpai)

Crossrefs

Cf. A000578, A062703, A069495, A075191.

Programs

Maple

a:= proc(n) option remember; local k, kk, p, q;
      for k from 1 +`if`(n=1, 1, iroot(a(n-1), 3))
      do kk:= k^3; p, q:= prevprime(kk), nextprime(kk);
         if (p+q)/2=kk then return kk fi
      od
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 21 2013

Mathematica

Select[Mean/@Partition[Prime[Range[1500000]],2,1],IntegerQ[Surd[#,3]]&] (* Harvey P. Dale, Oct 08 2014 *)
Select[Range[300]^3,#==Mean[{NextPrime[#],NextPrime[#,-1]}]&] (* Harvey P. Dale, Sep 02 2015 *)

PARI

is(n)=nextprime(n)+precprime(n)==2*n && ispower(n,3)
for(n=8,1e4,if(is(n^3), print1(n^3", "))) \\ Charles R Greathouse IV, Aug 25 2014

Formula

a(n) = A075191(n)^3.