cp's OEIS Frontend

A214840 Averages y of twin prime pairs that satisfy y = x^2 + x - 2.

%I A214840 #47 Aug 18 2024 11:27:29
%S A214840 4,18,108,180,270,810,4158,4968,5850,7308,10710,13338,17028,26730,
%T A214840 32940,38610,70488,72090,102078,117990,122148,128520,132858,153270,
%U A214840 228960,231840,240588,246510,249498,296478,326610,372708,391248,417960,429678,449568,453600
%N A214840 Averages y of twin prime pairs that satisfy y = x^2 + x - 2.
%C A214840 The above equation is one of a family of twin prime average-generating quadratics y = x^2 + x - c, where c can be any even integer not of the form 6d + 4.
%C A214840 For f(x) = x^2 + x - c,  f(-x) = f(x-1).
%C A214840 If c = 0, the positive x that satisfy y = x^2 + x - c are A088485.
%H A214840 Michael G. Kaarhus and Charles R Greathouse IV, <a href="/A214840/b214840.txt">Table of n, a(n) for n = 1..10000</a> (first 334 terms from Kaarhus)
%H A214840 M. G. Kaarhus, <a href="http://www.christaboveme.com/pri/quad-family.pdf">A Family of Twin Prime Quads</a> (PDF)
%e A214840 x =  2,  x =  4,  x = 10,  x = 13,  x = 16
%e A214840 x = 28,  x = 64,  x = 70,  x = 76,  x = 85
%t A214840 s = {4}; Do[If[PrimeQ[n - 1] && PrimeQ[n + 1] && IntegerQ[Sqrt[9 + 4 n]], AppendTo[s, n]], {n, 18, 453600, 6}]; s (* _Zak Seidov_, Mar 21 2013 *)
%t A214840 Select[Mean/@Select[Partition[Prime[Range[100000]],2,1],#[[2]]-#[[1]]==2&],IntegerQ[ Sqrt[ 9+4#]]&] (* _Harvey P. Dale_, Aug 18 2024 *)
%o A214840 (PARI) p=2;forprime(q=3,1e6,if(q-p>2,p=q;next);n=sqrtint(y=(p+q)\2);if(n^2+n-2==y,print1(y", "));p=q) \\ _Charles R Greathouse IV_, Mar 20 2013
%o A214840 (PARI) test(y)=if(isprime(y-1)&&isprime(y+1),print1(", "y))
%o A214840 print1(4);for(n=0,100,test(18*n*(2*n+1));test(18*(2*n^2+3*n+1))) \\ _Charles R Greathouse IV_, Mar 20 2013
%Y A214840 Subsequence of A014574. Cf. A088485.
%K A214840 nonn
%O A214840 1,1
%A A214840 _Michael G. Kaarhus_, Mar 07 2013