A088485 -id:A088485 - cp's OEIS frontend

A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

5, 7, 11, 13, 29, 31, 41, 43, 71, 73, 239, 241, 419, 421, 461, 463, 599, 601, 1481, 1483, 1721, 1723, 2549, 2551, 2969, 2971, 3539, 3541, 4421, 4423, 8009, 8011, 10301, 10303, 17291, 17293, 19181, 19183, 20021, 20023, 23561, 23563, 24179, 24181, 27059, 27061, 31151, 31153, 35531, 35533

Offset: 1

Bill McEachen, Nov 29 2015

This is a subset of A002327 and A002383 taken together. Note that 3 is not a member, as the pairing (3, 5) is excluded as defined, as 3 and 5 associate to different centers.
The corresponding n are in A088485.
The average of each twin prime pair is an oblong number (A002378). - Michel Marcus, Feb 04 2017

			For k = 6, k^2 + k = 6^2 + 6 = 42, and (41,43) is a twin prime pair, so 41 and 43 are in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3594 from G. C. Greubel)

Cf. A002327, A002378, A002383, A088485, A088486, A306889.

&cat[[n^2+n-1, n^2+n+1]: n in [0..250]| IsPrime(n^2+n-1) and IsPrime(n^2+n+1)]; // Vincenzo Librandi, Feb 05 2017

{#^2 + # - 1, #^2 + # + 1} & /@ Select[Range@ 200, PrimeQ[#^2 + # - 1] && PrimeQ[#^2 + # + 1] &] // Flatten (* Michael De Vlieger, Nov 30 2015 *)
Flatten[Select[Table[n^2 + n + {-1, 1}, {n, 0, 200}], And@@PrimeQ[#] &]] (* Vincenzo Librandi, Feb 05 2017 *)

PARI
```
genit()={my(maxx=1000);n=0;while(n
				
```

a(2n-1) = A088486(n). a(2n)=2+a(2n-1).

A088498 Numbers k such that k^2 + k - 1 and k^2 + k + 1 are twin primes and (k + 1)(k + 1) + k + 1 - 1 and (k + 1)(k + 1) + k + 1 + 1 are also twin primes.

Original entry on oeis.org

2, 5, 20, 455, 1364, 2204, 2450, 2729, 8540, 18485, 32198, 32318, 32780, 45863, 61214, 72554, 72560, 82145, 83258, 86603, 91370, 95198, 125333, 149330, 176888, 182909, 185534, 210845, 225665, 226253, 288419, 343160, 350090, 403940, 411500

Offset: 1

Pierre CAMI, Nov 11 2003

			20 is a term since 20^2 + 20 - 1 = 419, 419 and 421 are twin primes, 21^2 + 21 - 1 = 461, and 461 and 463 are also twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A088485.

Select[ Range[510397], PrimeQ[ #^2 + # - 1] && PrimeQ[ #^2 + # + 1] && PrimeQ[ #^2 + 3# + 1] && PrimeQ[ #^2 + 3# + 3] & ]
 Select[Range[412000],AllTrue[Flatten[{#^2+#+{1,-1},(#+1)(#+1)+#+{0,2}}], PrimeQ]&] (* Harvey P. Dale, Feb 12 2022 *)

Corrected and extended by Ray Chandler and Robert G. Wilson v, Nov 12 2003

A161866 Numbers k such that k^2+k+7 and k^2+k-7 are both prime.

Original entry on oeis.org

3, 5, 9, 12, 24, 29, 32, 39, 44, 50, 57, 59, 65, 102, 135, 137, 144, 170, 180, 207, 260, 267, 297, 302, 305, 344, 347, 360, 365, 369, 389, 404, 429, 464, 474, 495, 540, 555, 570, 612, 620, 659, 662, 689, 767, 774, 792, 824, 837, 872, 885, 900, 950, 954, 989

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 20 2009

			a(1)=3 as 12+-7 are primes. a(2)=5 as 30+-7 are primes.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A088485, A161863, A161864, A153417.

[k:k in [1..1000]| IsPrime(k^2+k+7) and IsPrime(k^2+k-7)]; // Marius A. Burtea, Feb 17 2020

q=7;lst7={};Do[p=n^2+n;If[PrimeQ[p-q]&&PrimeQ[p+q],AppendTo[lst7,n]], {n,0,7!}];lst7
Select[Range[1000],AllTrue[#^2+#+{7,-7},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 26 2021 *)

Definition rephrased by R. J. Mathar, Jun 23 2009

A173297 Numbers k such that exactly one of k^2 + k - 1 and k^2 + k + 1 is prime.

Original entry on oeis.org

1, 4, 9, 10, 11, 12, 13, 14, 16, 17, 19, 26, 27, 28, 30, 31, 33, 35, 39, 44, 45, 46, 48, 53, 55, 56, 57, 60, 62, 64, 65, 68, 69, 70, 71, 75, 76, 77, 78, 80, 83, 85, 86, 90, 93, 94, 96, 99, 100, 103, 105, 110, 111, 114, 115, 117, 119, 120, 125, 126, 130, 134, 140, 143, 144

Offset: 1

Juri-Stepan Gerasimov, Feb 15 2010

Numbers k such that either k-th oblong number-+1 is prime.

Indices k such that A002378(k)+1 or A002378(k)-1 is prime, but not both. -R. J. Mathar, Feb 21 2010

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002378, A088485.

isA173297 := proc(n) local p,pplus,pmin ; pmin := isprime(n*(1+n)-1) ; pplus := isprime(n*(1+n)+1) ; if pmin <> pplus then return true; else return false; end if; end proc: for n from 1 to 300 do if isA173297(n) then printf("%d,",n) ; end if; end do ; # R. J. Mathar, Feb 21 2010

46 and 86 inserted by R. J. Mathar, Feb 21 2010

Edited by Charles R Greathouse IV, Mar 24 2010

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

Original entry on oeis.org

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

Michael G. Kaarhus, Mar 07 2013

nonn

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.

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

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)

Subsequence of A014574. Cf. A088485.

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 *)

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

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

A286319 Prime p such that p^2-p-1 or p^2+p-1 is the smallest prime of a twin prime pair.

Original entry on oeis.org

2, 3, 5, 7, 41, 59, 67, 89, 101, 131, 139, 379, 457, 743, 761, 1193, 1201, 1381, 1549, 1567, 1747, 1789, 2137, 2411, 2557, 2647, 2663, 2729, 2731, 3011, 3221, 3251, 3449, 4057, 4159, 4447, 4561, 4751, 5179, 5641, 6173, 6397, 6599, 6833, 7229, 8669, 9059, 9157, 9323

Offset: 1

Pierre CAMI, May 11 2017

nonn

Union of A088483 and A120364.
3 is the only prime such that p^2-p-1 and p^2+p-1 are both the smallest of a prime twin pair.
For prime p > 3 if p+1 is divisible by 6 then the smallest prime of the prime twin pair is p^2+p-1 and p^2-p-1 if not.

			2^2+2-1=5 and (5,7) is a twin prime pair so a(1)=2.
3^2-3-1=5, 3^2+3-1=11 and (5,7), (11,13) are twin prime pairs so a(2)=3.
5^2+5-1=29 and (29,31) is a twin prime pair so a(3)=5.
7^2-7-1=41 and (41,43) is a twin prime pair so a(4)=7.

Pierre CAMI, Table of n, a(n) for n = 1..50000

Cf. A001359, A088483, A088485, A120364.

sptppQ[n_]:=AllTrue[{n^2-n-1,n^2-n+1},PrimeQ]||AllTrue[{n^2+n-1,n^2+ n+ 1},PrimeQ]; Select[Prime[Range[1200]],sptppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 04 2019 *)

A303550 Numbers k such that abs(60k^2 - 1710k + 12150) +- 1 are twin primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 27, 33, 34, 35, 36, 38, 41, 50, 56, 57, 64, 66, 69, 75, 81, 85, 86, 90, 93, 98, 103, 106, 119, 121, 133, 136, 141, 143, 146, 150, 181, 182, 189, 195, 202, 207, 208, 212, 215, 218, 219, 225

Offset: 1

Amiram Eldar, Apr 26 2018

The formula was discovered by Andrew T. Gazsi in 1961.
The polynomial can also be given as 30*(2*k - 27)*(k - 15). Its value is negative (-30) at k = 14 and 0 and k = 15.
Beiler erroneously claimed that the polynomial generates twin primes for k = 1 to 20.

			1 is in the sequence since 60*1^2 - 1710*1 + 12150 = 10500 and (10499, 10501) are twin primes.

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed., Dover Publications, Inc., New York, 1966, p. 225.
Joseph B. Dence and Thomas P. Dence, Elements of the Theory of Numbers, Academic Press, 1999, problem 1.94, p.35.
Andrew T. Gazsi, A Formula to Generate Prime Pairs, Recreational Mathematics Magazine, edited by Joseph S. Madachy, Issue 6, December 1961, p. 44.

Giovanni Resta, Table of n, a(n) for n = 1..10000
James Alston Hope Hunter and Joseph S. Madachy, Mathematical Diversions, D. van Nostrand Company, Inc., Princeton, New Jersey, 1963, p. 7.
Carlos Rivera, Problem 44. Twin-primes producing polynomials race, The Prime Puzzles & Problems Connection.

Cf. A001097, A001359, A006512, A088485, A108897, A124518, A124519, A139404, A185660.

filter:= proc(n) local k;
  k:= abs(60*n^2-1710*n+12150);
  isprime(k+1) and isprime(k-1)
end proc:
select(filter, [$1..300]); # Robert Israel, Jun 19 2018

f[n_] := 60n^2 - 1710n + 12150; aQ[n_]:=PrimeQ[f[n]-1] && PrimeQ[f[n]+1]; Select[Range[225], aQ]
Select[Range[250],AllTrue[Abs[60#^2-1710#+12150+{1,-1}],PrimeQ]&] (* Harvey P. Dale, May 17 2025 *)

f(n) = abs(60*n^2 - 1710*n + 12150);
isok(n) = my(fn=f(n)); isprime(fn-1) && isprime(fn+1); \\ Michel Marcus, Apr 27 2018

A248081 Least number k such that k^n + k +/- 1 are twin primes, or 0 if no such k exists.

Original entry on oeis.org

2, 2, 3, 2, 0, 8, 462, 0, 15, 3, 0, 30, 1500, 0, 30, 2, 0, 371, 11058, 0, 11289, 1599, 0, 4139, 44994, 0, 36951, 54, 0, 651, 2088, 0, 8841, 13061, 0, 81, 83406, 0, 20451, 3291, 0, 1821, 1128, 0, 122070, 20534, 0, 57852, 82875, 0, 15150, 132, 0, 515, 7215, 0, 37470, 2375, 0, 2340

Offset: 1

Derek Orr, Sep 30 2014

nonn

For n > 2, if n == 2 (mod 3), then k^n + k + 1 is divisible by k^2 + k + 1. Thus it will never be prime.

Cf. A088485, A236526, A236760.

a(n)=if(n>2&&n==Mod(2,3),return(0));k=1;while(!ispseudoprime(k^n+k-1)||!ispseudoprime(k^n+k+1),k++);k
n=1;while(n<50,print1(a(n),", ");n++)

cp's OEIS Frontend

A265006 Twin prime pairs of the form (k^2 + k - 1, k^2 + k + 1).

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A088498 Numbers k such that k^2 + k - 1 and k^2 + k + 1 are twin primes and (k + 1)*(k + 1) + k + 1 - 1 and (k + 1)*(k + 1) + k + 1 + 1 are also twin primes.

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A161866 Numbers k such that k^2+k+7 and k^2+k-7 are both prime.

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A173297 Numbers k such that exactly one of k^2 + k - 1 and k^2 + k + 1 is prime.

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

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A286319 Prime p such that p^2-p-1 or p^2+p-1 is the smallest prime of a twin prime pair.

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A303550 Numbers k such that abs(60*k^2 - 1710*k + 12150) +- 1 are twin primes.

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A088498 Numbers k such that k^2 + k - 1 and k^2 + k + 1 are twin primes and (k + 1)(k + 1) + k + 1 - 1 and (k + 1)(k + 1) + k + 1 + 1 are also twin primes.

A303550 Numbers k such that abs(60k^2 - 1710k + 12150) +- 1 are twin primes.