|
|
A238701
|
|
Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.
|
|
4
|
|
|
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 6, 5, 5, 5, 3, 4, 6, 6, 7, 6, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 4, 4, 6, 6, 4, 5, 5, 5, 7, 6, 6, 6, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 3, 4, 5, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,11
|
|
COMMENTS
|
Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n with q = floor((n-p)/m) and q^2 - 2 both prime.
In the case m = 1, this is a refinement of Goldbach's conjecture. In the case m = 2, this is stronger than Lemoine's conjecture (cf. A046927). The conjecture for m > 2 seems completely new. We view the conjecture as a natural extension of Goldbach's conjecture.
|
|
LINKS
|
|
|
EXAMPLE
|
a(11) = 2 since 2, floor((11-2)/4)= 2 and 2^2 - 2 are all prime, and 3, floor((11-3)/4) = 2 and 2^2 - 2 are all prime.
|
|
MATHEMATICA
|
PQ[n_]:=PrimeQ[n]&&PrimeQ[n^2-2]
p[n_, k_]:=PQ[Floor[(n-Prime[k])/4]]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 80}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|