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A214840
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Averages y of twin prime pairs that satisfy y = x^2 + x - 2.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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x = 2, x = 4, x = 10, x = 13, x = 16
x = 28, x = 64, x = 70, x = 76, x = 85
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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