A088483 -id:A088483 - cp's OEIS frontend

A088485 Numbers n such that n^2 + n - 1 and n^2 + n + 1 are twin primes.

2, 3, 5, 6, 8, 15, 20, 21, 24, 38, 41, 50, 54, 59, 66, 89, 101, 131, 138, 141, 153, 155, 164, 176, 188, 203, 206, 209, 215, 218, 231, 236, 246, 288, 290, 309, 314, 351, 378, 395, 405, 453, 455, 456, 495, 500, 518, 530, 551, 560, 624, 644, 668, 686, 720, 728, 743, 761, 798, 825, 890, 915, 950, 974, 981

Offset: 1

Pierre CAMI, Nov 09 2003

A265006 gives these primes. - Derek Orr, Dec 24 2015

			20*20 + 20 - 1 = 419, 419 and 421 twin primes, 20 is the 7th of the sequence

Pierre CAMI, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A088483, A265006.

[n: n in [1..2*10^3] |IsPrime(n^2+n-1) and IsPrime(n^2+n+1)]; // Vincenzo Librandi, Dec 26 2015

Select[Range[500], PrimeQ[ #^2+#-1] && PrimeQ[ #^2+#+1] &] (* T. D. Noe, Jun 22 2004 *)
Select[Range[1000],AllTrue[#^2+#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 12 2017 *)

for(n=1,10^3,if(isprime(n^2+n-1)&&isprime(n^2+n+1),print1(n,", "))) \\ Derek Orr, Dec 24 2015

Corrected description from T. D. Noe, Jun 22 2004

A120364 Primes p such that p^2-p-1 and p^2-p+1 are twin primes.

Original entry on oeis.org

3, 7, 67, 139, 379, 457, 1201, 1381, 1549, 1567, 1747, 1789, 2137, 2557, 2647, 2731, 4057, 4159, 4447, 4561, 5179, 5641, 6397, 9157, 9661, 9829, 9967, 10369, 11467, 11677, 12487, 12781, 13339, 13399, 15241, 17299, 17977, 19207, 19417, 19429

Offset: 1

Pierre CAMI, Jun 26 2006

nonn

One more than the entries of (A006093 intersect A088485). - Danny Rorabaugh, May 15 2017

			3*3-3-1=5 3*3-3+1=7, 5 and 7 twin primes so a(1)=3;
5*5-5-1=19 5*5-5+1=21 composite;
7*7-7-1=41 7*7-7+1=43, 41 and 43 twin primes so a(2)=7.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A088483, A088485.

Select[Prime[Range[2500]], PrimeQ[ #^2 - # - 1] && PrimeQ[ #^2 - # + 1] &] (* Stefan Steinerberger, Jul 22 2006 *)
Select[Prime[Range[2500]],AllTrue[#^2-#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Jul 25 2021 *)

More terms from Stefan Steinerberger and Rick L. Shepherd, Jul 22 2006

A228968 Prime p such that p and p+2 are twin primes and p^2+p-1 p^2+p+1 are also twin primes.

Original entry on oeis.org

3, 5, 41, 59, 101, 2729, 3251, 9719, 11549, 12251, 19211, 28619, 41201, 47711, 49391, 55439, 58229, 61979, 63029, 63311, 79631, 81371, 85331, 103391, 122039, 135719, 153509, 157349, 164249, 167441, 178601, 188861, 197711, 208001, 209819, 216779, 219311, 226451

Offset: 1

Pierre CAMI, Sep 10 2013

nonn

Subsequence of A088483.

			3 and 5 twin primes as 3*3+3-1=11 and 13, a(2)=3.
5 and 7 twin primes as 5*5+5-1=29 and 31, a(3)=5.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A088483.

Select[Transpose[Select[Partition[Prime[Range[21000]],2,1],#[[2]]-#[[1]] == 2&]][[1]],AllTrue[ #^2+#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 10 2014 *)

is(n)=isprime(n)&&isprime(n+2)&&isprime(n^2+n-1)&&isprime(n^2+n+1) \\ Charles R Greathouse IV, Sep 10 2013

ABC2 $a & $a+2 & $a^2+$a-1 & $a^2+$a+1
a: 1 to 3000000
Charles R Greathouse IV, Sep 10 2013

[x for x in primes_first_n(900) if x+2 in Primes() and x^2+x-1 in Primes() and x^2+x+1 in Primes()] #Tom Edgar, Sep 10 2013

A158295 Primes p such that p^3-p-+1 are twin primes.

Original entry on oeis.org

2, 11, 31, 41, 239, 521, 2309, 4099, 4409, 4441, 4651, 5009, 5039, 5261, 6481, 6871, 7129, 8609, 9391, 10259, 12841, 13759, 14519, 14879, 14939, 15569, 16871, 18451, 20369, 22441, 24049, 25841, 28151, 28279, 29429, 30181, 30631, 32089, 32299, 36781

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 15 2009

nonn

Primes p such that p^3+p-+1 are twin primes, so far only one: 3. 3^3+3=30-+1 = primes.

Primes in the sequence A236524. Odd primes are congruent to either 1 mod 10 or 9 mod 10. - Derek Orr, Jan 27 2014

			2^3-2=6-+1 = 5,7 primes, 11^3-11-+1 = 1319,1321 primes...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A120364, A088483, A236524, A236477, A126421.

lst={};Do[p=Prime[n];a=p^3-p;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p]],{n,8!}];lst
Select[Prime[Range[3500]],And@@PrimeQ[#^3-#+{1,-1}]&] (* Harvey P. Dale, Jan 05 2013 *)

s=[]; forprime(p=2, 40000, if(isprime(p^3-p-1) && isprime(p^3-p+1), s=concat(s, p))); s /* Colin Barker, Jan 28 2014 */

import sympy
from sympy import isprime
{print(p) for p in range(10**5) if isprime(p) and isprime(p**3-p-1) and isprime(p**3-p+1)} # Derek Orr, Jan 27 2014

A088484 Sequence of the primes P = p(k)^2 + p(k) - 1 such that P and P + 2 are twin primes where p(k) denotes k-th prime.

Original entry on oeis.org

5, 11, 29, 1721, 3539, 8009, 10301, 17291, 552791, 579881, 1424441, 5815331, 7094231, 7450169, 9069131, 10378061, 10572251, 11899049, 22576751, 38112101, 43553399, 46696721, 52265669, 75160229, 82074539, 86927651, 90658961, 94468679, 94935791, 103052951, 116240741

Offset: 1

Pierre CAMI, Nov 09 2003

nonn

			For k = 26, p(26) = 101, 101^2 + 101 - 1 = 10301, 10301 and 10303 twin primes, therefore 10301 is a term.

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

p(k) given in A088483.

f[n_] := n^2 + n - 1; f /@ Select[Range[10^4], And @@ PrimeQ[{#, f[#], f[#] + 2}] &] (* Amiram Eldar, Dec 27 2019 *)

Corrected by T. D. Noe, Nov 15 2006

More terms from Amiram Eldar, Dec 27 2019

A155185 Primes in A155175.

Original entry on oeis.org

5, 13, 113, 1741, 5101, 8581, 9941, 21841, 26681, 47741, 82013, 481181, 501001, 1009621, 2356621, 2542513, 3279361, 3723721, 4277813, 7757861, 8124481, 13204661, 25311613, 30772013, 44170601, 48619661, 51521401, 52541501, 54236113, 60731221, 72902813

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Hypotenuse C (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes. p=1,q=2,a=3,b=4,c=5=prime,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[c],AppendTo[lst,c]]],{n,8!}];lst (* corrected by Ray Chandler, Feb 11 2020 *)

Sequence corrected by Ray Chandler, Feb 11 2020

A158296 Primes p such that 12*p^2 -+ 1 are twin primes.

Original entry on oeis.org

3, 11, 13, 17, 29, 227, 283, 491, 647, 739, 953, 1151, 1471, 1511, 1879, 1889, 2129, 2251, 2297, 2593, 2633, 3347, 3539, 3559, 3643, 3877, 3919, 4231, 4327, 4547, 4673, 4801, 4999, 5051, 6451, 6653, 6737, 6779, 6983, 7741, 7937, 8179, 8219, 8231, 8389

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 15 2009

nonn

			3 is in the sequence since 12*3^2 = 108 and (107, 109) are twin primes.

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

Cf. A120364, A088483, A158295.

lst={};Do[p=Prime[n];a=12*p^2;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1200]],AllTrue[12#^2+{1,-1},PrimeQ]&] (* Harvey P. Dale, Sep 15 2021 *)

A155186 Primes in A155171.

Original entry on oeis.org

2, 7, 29, 101, 107, 197, 227, 457, 647, 829, 1549, 1627, 2221, 2309, 2347, 2521, 2677, 2801, 3181, 3299, 3529, 3541, 3557, 3739, 3769, 4231, 4549, 4871, 4987, 5651, 5827, 5881, 6037, 6079, 6637, 6827, 7517, 7639, 7937, 9787, 11621, 12041, 12329, 13009

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Numbers p (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[p],AppendTo[lst,p]]],{n,8!}];lst

A158297 Primes p such that 12*p^3+-1 are twin primes.

Original entry on oeis.org

11, 239, 449, 619, 2099, 2711, 3109, 3889, 4591, 5519, 8539, 9719, 12071, 17981, 19441, 21569, 28949, 29399, 32771, 38189, 38201, 40709, 41771, 44699, 45949, 47149, 50741, 52189, 52379, 52501, 52639, 55339, 56249, 58831, 61561, 62861

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 15 2009

nonn

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

Cf. A120364, A088483, A158295, A158296.

[p:p in PrimesUpTo(63000)| IsPrime(12*p^3-1) and NextPrime(12*p^3-1) eq 12*p^3+1]; // Marius A. Burtea, Jan 23 2020

lst={};Do[p=Prime[n];a=12*p^3;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p]],{n,8!}];lst
Select[Prime[Range[7000]],AllTrue[12#^3+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 03 2019 *)

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 *)

cp's OEIS Frontend

A088485 Numbers n such that n^2 + n - 1 and n^2 + n + 1 are twin primes.

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A120364 Primes p such that p^2-p-1 and p^2-p+1 are twin primes.

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A228968 Prime p such that p and p+2 are twin primes and p^2+p-1 p^2+p+1 are also twin primes.

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A158295 Primes p such that p^3-p-+1 are twin primes.

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A088484 Sequence of the primes P = p(k)^2 + p(k) - 1 such that P and P + 2 are twin primes where p(k) denotes k-th prime.

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A155185 Primes in A155175.

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A158296 Primes p such that 12*p^2 -+ 1 are twin primes.

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A155186 Primes in A155171.

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