A224794 Smallest prime p such that 2n+1 = 2p - q^3 for some odd prime q, or 0 if no such prime exists.

173, 2459, 17, 67, 19, 113497, 179, 71, 23, 73, 677, 25339, 0, 74453, 29, 79, 31, 1117, 191, 83, 193, 25349, 37, 1123, 197, 89, 41, 3594557, 43, 1129, 3461, 12227, 47, 97, 701, 647551, 3467, 101, 53, 103, 1124087, 647557, 709, 107, 59, 109, 61, 113539, 6133

Offset

1,1

Comments

See the odd numbers of the form p^3 - 2q where p and q are primes (A224730, A185046).

a(n) = 0 for n = 13, 62, 171, 364, 665, 1098, ... and 2n+1 = 27, 125, 343, 729, ... are the odd cubes 3^3, 5^3, 7^3, ...

The corresponding primes q are in A224795.

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

Remark: if m = c^3 for any odd integer c, then c^3 = 2p - q^3 is impossible because c^3 + q^3 = 2p => (c+q)(c^2 - cq + q^2) with c+q even of the form c+q = 2a => p = a(c^2 - cq + q^2) absurd because p is prime.

Links

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

Example

a(4) = 67 because, for (p, q) = (67, 5), 2*4 + 1 = 9 = 2*67 - 5^3 = 134 - 125 = 9.

Maple

for n from 3 by 2 to 200 do:jj:=0:for j from 1 to 50000 while (jj=0) do:q:=ithprime(j):p:=(q^3+n)/2:if type(p, prime)=true  then jj:=1: printf(`%d, `, p):else fi:od:if jj=0 then printf(`%d, `, 0):else fi:od:

Crossrefs

Cf. A185046, A224730, A224795.

Sequence in context: A142134 A142849 A271978 * A209809 A185707 A185701

Adjacent sequences: A224791 A224792 A224793 * A224795 A224796 A224797

Keyword

nonn

Author

Michel Lagneau, Apr 18 2013

Status

approved