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

4, 18, 108, 180, 270, 810, 4158, 4968, 5850, 7308, 10710, 13338, 17028, 26730, 32940, 38610, 70488, 72090, 102078, 117990, 122148, 128520, 132858, 153270, 228960, 231840, 240588, 246510, 249498, 296478, 326610, 372708, 391248, 417960, 429678, 449568, 453600

Offset: 1

Author

Michael G. Kaarhus, Mar 07 2013

Keywords

• nonn

Comments

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.

For f(x) = x^2 + x - c, f(-x) = f(x-1).

If c = 0, the positive x that satisfy y = x^2 + x - c are A088485.

Examples

			x =  2,  x =  4,  x = 10,  x = 13,  x = 16
x = 28,  x = 64,  x = 70,  x = 76,  x = 85
		

Links

• Michael G. Kaarhus and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 334 terms from Kaarhus)
• M. G. Kaarhus, A Family of Twin Prime Quads (PDF)

Crossrefs

Subsequence of A014574. Cf. A088485.

Programs

Mathematica

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 *)
Select[Mean/@Select[Partition[Prime[Range[100000]],2,1],#[[2]]-#[[1]]==2&],IntegerQ[ Sqrt[ 9+4#]]&] (* Harvey P. Dale, Aug 18 2024 *)

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

PARI

test(y)=if(isprime(y-1)&&isprime(y+1),print1(", "y))
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