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

1,1

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

Links

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

Example

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

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[#]]&]

Prog

(PARI) for(n=1, 125, n%2==1&&!isprime(n^3+2)&&print1(n^3, ", "))
(Magma) [n^3: n in [1..150 by 2] | not IsPrime(n^3+2)]; // Vincenzo Librandi, Feb 28 2016

Crossrefs

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

Sequence in context: A373590 A043395 A336354 * A202852 A117689 A178374

Adjacent sequences: A269343 A269344 A269345 * A269347 A269348 A269349

Keyword

nonn

Author

Waldemar Puszkarz, Feb 24 2016

Status

approved