login
A224481
Positive integers x such that x^2 - 34 is the average of a twin prime pair.
1
8, 26, 46, 58, 74, 76, 82, 92, 134, 164, 236, 248, 304, 314, 362, 368, 394, 416, 454, 496, 502, 512, 544, 568, 592, 598, 632, 668, 706, 734, 772, 776, 788, 818, 824, 844, 898, 944, 986, 1142, 1184, 1324, 1328, 1346, 1426, 1436, 1462, 1502, 1522, 1612, 1766
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
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
Cf. A080149.
Sequence in context: A271989 A069952 A031085 * A031307 A122854 A138502
KEYWORD
nonn
AUTHOR
Michael G. Kaarhus, Apr 09 2013
STATUS
approved