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

343, 729, 1331, 2197, 3375, 4913, 6859, 9261, 12167, 15625, 19683, 29791, 35937, 42875, 50653, 59319, 68921, 79507, 103823, 117649, 132651, 148877, 166375, 185193, 205379, 226981, 300763, 389017, 421875, 456533, 493039, 531441, 614125, 658503, 704969

Offset: 1

Author

Waldemar Puszkarz, Feb 24 2016

Keywords

• nonn

Comments

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.

The cubes of this property are also the cubes in A269345.

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

Examples

			For n=1, 343 = 7^3 and 345 = 343+2 is a composite, so 343 is a term.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000578 (the cubes), A067200 (cube roots of terms that complement this sequence), A269345 (supersequence).

Programs

Magma

[n^3: n in [1..150 by 2] | not IsPrime(n^3+2)]; // Vincenzo Librandi, Feb 28 2016

Mathematica

Select[Range[1,125,2]^3, !PrimeQ[#+2]&]
Select[Range[125]^3, !PrimeQ[#+2]&&OddQ[#]&]
Select[Select[Range[2000000], OddQ[#]&& !PrimeQ[#]&& !PrimeQ[#+2]&], IntegerQ[CubeRoot[#]]&]

PARI

for(n=1, 125, n%2==1&&!isprime(n^3+2)&&print1(n^3, ", "))