A155173 -id:A155173 - cp's OEIS frontend

A088483 Primes p such that p^2+p-1 and p^2+p+1 are twin primes.

2, 3, 5, 41, 59, 89, 101, 131, 743, 761, 1193, 2411, 2663, 2729, 3011, 3221, 3251, 3449, 4751, 6173, 6599, 6833, 7229, 8669, 9059, 9323, 9521, 9719, 9743, 10151, 10781, 11549, 11933, 12143, 12251, 12473, 12641, 13553, 13613, 14939, 15569, 16301

Offset: 1

Pierre CAMI, Nov 09 2003

nonn

Also primes in A155173 = Short leg A of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs... - Vladimir Joseph Stephan Orlovsky, Jan 21 2009
All terms >3 are congruent to 5 modulo 6. - Zak Seidov, Mar 21 2014
Intersection of A000040 and A088485. - Danny Rorabaugh, May 15 2017

			a(7) = 101: 101*101 + 101 - 1 = 10301, 10301 and 10303 twin primes.

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

Cf. A001097, A088484, A088485.

lst={};Do[p=Prime[n];If[PrimeQ[p1=p*p+p-1]&&PrimeQ[p1+2],AppendTo[lst,p]],{n,2*7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 21 2009 *)
Select[Prime[Range[2000]],AllTrue[#^2+#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Feb 22 2023 *)

forprime(p=2,1e5,if(isprime(p^2+p-1)&&isprime(p^2+p+1),print1(p", "))) \\ Charles R Greathouse IV, Dec 27 2011

A155174 Long leg B 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=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

4, 12, 112, 220, 840, 1740, 3960, 5100, 8580, 9940, 11704, 12012, 20604, 21840, 23112, 26680, 47740, 61600, 78012, 82012, 102604, 103512, 122512, 151800, 276024, 289560, 340312, 418612, 481180, 501000, 660100, 711624, 838512, 901824, 931612

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173

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

A155175 Hypotenuse C 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=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

5, 13, 113, 221, 841, 1741, 3961, 5101, 8581, 9941, 11705, 12013, 20605, 21841, 23113, 26681, 47741, 61601, 78013, 82013, 102605, 103513, 122513, 151801, 276025, 289561, 340313, 418613, 481181, 501001, 660101, 711625, 838513, 901825, 931613

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174

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

A155176 Perimeter s/6 (divided by 6) 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=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 5, 40, 77, 287, 590, 1335, 1717, 2882, 3337, 3927, 4030, 6902, 7315, 7740, 8932, 15965, 20592, 26070, 27405, 34277, 34580, 40920, 50692, 92132, 96647, 113575, 139690, 160557, 167167, 220225, 237407, 279720, 300832, 310765, 336777, 389895

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,s/6]],{n,8!}];lst

A155177 Area ar/6 (divided by 6) 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=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

1, 5, 140, 385, 2870, 8555, 29370, 42925, 93665, 116795, 149226, 155155, 348551, 380380, 414090, 513590, 1229305, 1801800, 2567895, 2767905, 3873301, 3924830, 5053620, 6970150, 17090486, 18362930, 23396450, 31919165, 39336465, 41791750

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5, ar=3*4/2=6, s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;ar=a*b/2;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,ar/6]],{n,8!}];lst

A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

Original entry on oeis.org

1, 7916, 35882, 37816, 47491, 128429, 131830, 146471, 154799, 157579, 170219, 174964, 187544, 207829, 208039, 222887, 223142, 262502, 291544, 319825, 327602, 331627, 353857, 476681, 477659, 494207, 522025, 537454, 540682, 558161, 571670

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes,pr=3*4*5=60-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177

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;pr=a*b*c;If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[pr-1]&&PrimeQ[pr+1],AppendTo[lst,n]],{n,3*9!}];lst

A155180 Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

Original entry on oeis.org

3, 15833, 71765, 75633, 94983, 256859, 263661, 292943, 309599, 315159, 340439, 349929, 375089, 415659, 416079, 445775, 446285, 525005, 583089, 639651, 655205, 663255, 707715, 953363, 955319, 988415, 1044051, 1074909, 1081365, 1116323

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes,pr=3*4*5=60-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178

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;pr=a*b*c;If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[pr-1]&&PrimeQ[pr+1],AppendTo[lst,a]],{n,3*9!}];lst

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

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

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 3, 11, 71, 227, 491, 683, 1103, 1187, 2591, 3923, 4271, 4931, 6737, 7193, 7703, 8093, 8753, 8963, 9173, 9377, 10271, 13043, 13451, 13997, 15233, 15443, 15803, 15887, 17957, 18701, 19961, 20681, 21701, 22031, 22073, 24371, 24473, 24683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1, q=2(prime), a=3, b=4, c=5, s=12-+1 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, A155186, A068231, A129517, A054574, A132281.

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[q],AppendTo[lst,q]]],{n,8!}];lst

cp's OEIS Frontend

A088483 Primes p such that p^2+p-1 and p^2+p+1 are twin primes.

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A155174 Long leg B 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.

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A155175 Hypotenuse C 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.

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A155176 Perimeter s/6 (divided by 6) 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.

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A155177 Area ar/6 (divided by 6) 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.

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A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides 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, pr=a*b*c, pr-+1 are primes.

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A155180 Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides 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, pr=a*b*c, pr-+1 are primes.

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A155185 Primes in A155175.

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A155186 Primes in A155171.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), 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.

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A155174 Long leg B 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=2pq, s=a+b+c, s-+1 are primes.

A155175 Hypotenuse C 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=2pq, s=a+b+c, s-+1 are primes.

A155176 Perimeter s/6 (divided by 6) 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=2pq, s=a+b+c, s-+1 are primes.

A155177 Area ar/6 (divided by 6) 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=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

A155178 Numbers p of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

A155180 Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.