For
r
≧
1
r \geqq 1
and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes
p
≦
N
p \leqq N
such that
p
+
2
r
p + 2r
is the least prime greater than p is asymptotic to
\[
∫
2
N
(
∑
k
=
1
r
A
r
,
k
(
log
x
)
k
+
1
)
d
x
,
\int _2^N {\left ( {\sum \limits _{k = 1}^r {\frac {{{A_{r,k}}}}{{{{(\log x)}^{k + 1}}}}} } \right )} \;dx,
\]
where the
A
r
,
k
{A_{r,k}}
are certain constants. We describe a method for computing these constants. Related constants are given to 10D for
r
=
1
(
1
)
40
r = 1(1)40
, and some empirical evidence supporting the conjecture is mentioned.