OFFSET
1,1
COMMENTS
y = x^2 - 34 is one of a family of quadratics y = x^2 + c that produces averages of twin prime pairs. The first 24 negative numbers c that produce averages are congruent to either 0 or 2 (mod 6) (as calculated by maxima), and they differ by no more than 6. Other than that, I have not found an order to the sequence of negative numbers c. The first 11 positive numbers c that produce averages are apparently the beginning of all integers >= 2 that are equivalent to {2,0,2,0...} (mod 6).
If c=2, then the x that satisfy y = x^2 + c are A080149.
Apparently, there are infinitely many numbers c that produce twin prime averages. Here are some of them: (-84, -78, -76, -72, -70, -66, -64, -60, -58, -54, -52, -46, -42, -40, -36, -34, -30, -28, -22, -18, -16, -12, -6, -4, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32).
Dickson's conjecture implies that this sequence is infinite. Bateman-Horn-Stemmler gives conjectured growth. - Charles R Greathouse IV, Apr 10 2013
LINKS
Michael G. Kaarhus, Table of n, a(n) for n = 1..10000
EXAMPLE
26 is in this sequence, because 26^2 - 34 = 642, which is the average of the twin prime pair (641, 643).
MATHEMATICA
nn=1000; av = Select[Prime[Range[PrimePi[nn^2]]], PrimeQ[# + 2] &] + 1; Select[Range[nn], MemberQ[av, #^2 - 34] &] (* T. D. Noe, Apr 09 2013 *)
nn = 2000; Select[Range[8, nn, 2], PrimeQ[p = #^2 - 35] && PrimeQ[p + 2] &] (* Zak Seidov, Apr 27 2013 *)
Select[Range[3, 1800], AllTrue[#^2-{35, 33}, PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2020 *)
PROG
(PARI) is(n)=isprime(n^2-35)&&isprime(n^2-33) \\ Charles R Greathouse IV, Apr 10 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael G. Kaarhus, Apr 09 2013
STATUS
approved