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

1,1

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000578, A062703, A069495, A075191.

Sequence in context: A316875 A317603 A269660 * A017115 A269203 A005609

Adjacent sequences: A234237 A234238 A234239 * A234241 A234242 A234243

Keyword

nonn

Author

K. D. Bajpai, Dec 21 2013

Status

approved