A224481 Positive integers x such that x^2 - 34 is the average of a twin prime pair.

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

Author

Michael G. Kaarhus, Apr 09 2013

Keywords

• nonn

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

Examples

			26 is in this sequence, because 26^2 - 34 = 642, which is the average of the twin prime pair (641, 643).
		

Links

• Michael G. Kaarhus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A080149.

Programs

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 *)

PARI

is(n)=isprime(n^2-35)&&isprime(n^2-33) \\ Charles R Greathouse IV, Apr 10 2013