A185046 Smallest prime p such that 2n+1 = p^3 - 2q for some odd prime q, or 0 if no such prime exists.

5, 3, 5, 7, 13, 3, 13, 3, 5, 3, 3, 11, 0, 7, 5, 19, 37, 11, 5, 7, 5, 7, 37, 11, 5, 31, 53, 31, 13, 23, 5, 7, 5, 7, 13, 23, 13, 19, 5, 7, 421, 47, 5, 7, 5, 11, 13, 11, 5, 43, 5, 11, 61, 23, 5, 19, 5, 7, 5, 5, 53, 7, 17, 7, 13, 11, 13, 7, 113, 7, 373, 11, 17, 7

Offset

1,1

Comments

a(n) = 0 for n = 13, 171, 364, 1098, 2456, 3429, 6083, 7812, 9841, 12194, 14895, 17968,... and 2n+1 = 27, 343, 729,... is a class of cubes.

The corresponding primes q are in A224730.

Conjecture: The odd numbers different from a cube are of the form m = p^3 - 2q where p and q are prime numbers.

Remark: Its converse is false: there exists cubes m = c^3 that are in the sequence with the form c^3 = p^3 - 2q, where p-c = 2, and q of the form x^2 +x*y+y^2 (see A007645). For example: 5^3 = 7^3 - 2*109.

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Example

a(4) = 7 because, for (p, q) = (7, 167) => 2*4+1 = 9 = 7^3 - 2*167 = 343 - 334 = 9.

Maple

for n from 3 by 2 to 200 do:
      jj:=0:
          for j from 1 to 10000 while (jj=0) do:
             p:=ithprime(j):q:=(p^3-n)/2:
             if q> 0 and type(q, prime)=true
             then
             jj:=1:printf(`%d, `, p):
             else
             fi:
         od:
            if jj=0 then
            printf(`%d, `, 0):
            else
            fi:
     od:

Crossrefs

Cf. A007645, A224730.

Sequence in context: A336055 A131925 A128010 * A114740 A330523 A361546

Adjacent sequences: A185043 A185044 A185045 * A185047 A185048 A185049

Keyword

nonn

Author

Michel Lagneau, Apr 17 2013

Status

approved