cp's OEIS Frontend

A269346 Perfect cubes that are not the difference of two primes.

%I A269346 #40 Sep 08 2022 08:46:15
%S A269346 343,729,1331,2197,3375,4913,6859,9261,12167,15625,19683,29791,35937,
%T A269346 42875,50653,59319,68921,79507,103823,117649,132651,148877,166375,
%U A269346 185193,205379,226981,300763,389017,421875,456533,493039,531441,614125,658503,704969
%N A269346 Perfect cubes that are not the difference of two primes.
%C A269346 An even number can be the difference of two primes, but an odd one can only be if an odd number m is such that m+2 is prime. Since a(n) is odd and such that a(n)+2 is composite, a(n) cannot be such a difference.
%C A269346 The cubes of this property are also the cubes in A269345.
%C A269346 It is still an open conjecture that every even number is the difference of 2 primes. On the other hand, a computer test shows that all even cubes <= 10^21 can be written as the difference of 2 primes. The computer program generating the sequence needs an additional part to test for even cubes besides checking that for odd m^3, m^3+2 is composite. - _Chai Wah Wu_, Mar 03 2016
%H A269346 Chai Wah Wu, <a href="/A269346/b269346.txt">Table of n, a(n) for n = 1..10000</a>
%e A269346 For n=1, 343 = 7^3 and 345 = 343+2 is a composite, so 343 is a term.
%t A269346 Select[Range[1,125,2]^3, !PrimeQ[#+2]&]
%t A269346 Select[Range[125]^3, !PrimeQ[#+2]&&OddQ[#]&]
%t A269346 Select[Select[Range[2000000], OddQ[#]&& !PrimeQ[#]&& !PrimeQ[#+2]&], IntegerQ[CubeRoot[#]]&]
%o A269346 (PARI) for(n=1, 125, n%2==1&&!isprime(n^3+2)&&print1(n^3, ", "))
%o A269346 (Magma) [n^3: n in [1..150 by 2] | not IsPrime(n^3+2)]; // _Vincenzo Librandi_, Feb 28 2016
%Y A269346 Cf. A000578 (the cubes), A067200 (cube roots of terms that complement this sequence), A269345 (supersequence).
%K A269346 nonn
%O A269346 1,1
%A A269346 _Waldemar Puszkarz_, Feb 24 2016