A088485 -id:A088485 - cp's OEIS frontend

A088497 Duplicate of A088485.

2, 3, 5, 6, 8, 15, 20, 21, 24, 38, 41, 50, 54, 59, 66, 89, 101, 131, 138, 141, 153, 155, 164

Offset: 1

dead

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

A108309 Consider the triangle of odd numbers where the n-th row has the next n odd numbers. The sequence is the number of primes in the n-th row.

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 3, 4, 4, 5, 3, 4, 6, 4, 6, 6, 4, 6, 7, 6, 8, 7, 5, 8, 9, 8, 7, 8, 9, 8, 9, 10, 10, 8, 10, 12, 5, 12, 12, 13, 9, 11, 11, 9, 13, 14, 9, 14, 14, 10, 10, 19, 14, 12, 12, 12, 12, 16, 15, 16, 15, 13, 18, 16, 16, 12, 16, 17, 15, 16, 18, 14, 15, 20, 18, 19, 14, 19, 20, 18, 16

Offset: 1

Giovanni Teofilatto, Jul 25 2005

Except for the initial term, a(n)>=2 because in the interval 2n-1 of odd numbers there are always at least two primes.
For n>2, this is the same as the number of primes between n^2-n and n^2+n, which is the sum of A089610 and A094189. - T. D. Noe, Sep 16 2008
a(n) = SUM(A010051(A176271(n,k)): 1<=k<=n). - Reinhard Zumkeller, Apr 13 2010
From Pierre CAMI, Sep 03 2014: (Start)
For n>1 a(n)~floor(1/2 + n/log(n)).
The number of primes < n^2 is ~ n^2/2/log(n) by the prime number theorem, as a(n) ~ floor(1/2 + n/log(n)) we have:
n^2/2/log(n) ~ 1 + floor(1/2 + 2/log(2)) + floor(1/2 + 3/log(3)) + floor(1/2 + 4/log(4)) + ... + floor(1/2 + (n-1)/log(n-1)) + floor(1/2 + n/log(n)).
For n=16000 the number of primes < n^2 is 13991985, the sum: 1 + floor(1/2 + 2/log(2)) + floor(1/2 + 3/log(3)) + floor(1/2 + 4/log(4))+ ... + floor(1/2 + (n-1)/log(n-1)) + floor(1/2 + n/log(n)) is 13991101 and (n^2)/(2*log(n)) is 13222671.
So between n^2+n and n^2+3*n there are n odd numbers and ~floor(1/2 + n/log(n)) prime numbers.
The twin primes are of the form T1=n^2+n-1 and T2=n^2+n+1, or n^2+n+T1 and n^2+n+T2 with T1<=2*n-1, or n^2+n+P and n^2+n+P(-2 or +2) with P prime <=2*n-1.
(End)

			Triangle begins:
1: 1 -> 0 primes,
2: 3,5 -> 2 primes,
3: 7,9,11 -> 2 primes,
4: 13,15,17,19 -> 3 primes.

T. D. Noe, Table of n, a(n) for n=1..10000
Pierre CAMI, Table of n, a(n) and Floor(1/2+n/log(n))for n=1..10000

Cf. A014085, A088485.

a108309 = sum . (map a010051) . a176271_row
-- Reinhard Zumkeller, May 24 2012

seq(numtheory:-pi(n^2+n-1)-numtheory:-pi(n^2-n),n=1..100); # Robert Israel, Sep 03 2014

f[n_] := PrimePi[n^2 + n - 1] - PrimePi[n^2 - n]; Table[f[n], {n, 81}] (* Ray Chandler, Jul 26 2005 *)

Edited and extended by Ray Chandler, Jul 26 2005

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

A088486 Primes p of the form k*(k + 1) - 1 such that p and p + 2 are twin primes.

Original entry on oeis.org

5, 11, 29, 41, 71, 239, 419, 461, 599, 1481, 1721, 2549, 2969, 3539, 4421, 8009, 10301, 17291, 19181, 20021, 23561, 24179, 27059, 31151, 35531, 41411, 42641, 43889, 46439, 47741, 53591, 55931, 60761, 83231, 84389, 95789, 98909, 123551, 143261, 156419, 164429

Offset: 1

Pierre CAMI, Nov 09 2003

nonn

a(n) + 1 is an oblong number (A002378). - Michel Marcus, Feb 04 2017

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

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

Cf. A002378, A088485.

Select[Table[k(k+1)-1,{k,500}],AllTrue[#+{0,2},PrimeQ]&] (* Harvey P. Dale, Jul 08 2024 *)

a(n) = A088485(n)*(A088485(n) + 1) - 1. - R. J. Mathar, Nov 14 2009

Sign in definition corrected by R. J. Mathar, Nov 13 2009

A138303 Numbers k such that k(k+1)-1 and k(k+1)+1 are twin primes and k(k+3)-1 and k(k+3)+1 are also twin primes.

Original entry on oeis.org

3, 15, 495, 624, 1308, 1365, 1494, 1659, 1848, 6363, 8064, 12780, 13908, 14013, 16530, 18018, 18654, 28575, 30885, 31020, 38130, 43674, 48735, 50379, 57198, 60915, 61059, 62943, 75768, 76359, 76569, 80949, 81075, 83028, 84045, 86085, 88185, 104664, 105588

Offset: 1

Pierre CAMI, Mar 13 2008

nonn

			3*4=12: 11 and 13 are twin primes; 3*6=18 and 17 and 19 are also twin primes.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 500 terms from Pierre CAMI)

Subsequence of A088485.

Select[Range[106000],AllTrue[{#*(#+1)-1,#*(#+1)+1,#*(#+3)-1,#*(#+3)+1},PrimeQ]&] (* James C. McMahon, Jun 06 2025 *)

isok(n) = isprime(n*(n+1)-1) && isprime(n*(n+1)+1) && isprime(n*(n+3)-1) &&  isprime(n*(n+3)+1); \\ Michel Marcus, Sep 23 2019

A161863 Numbers k such that k^2+k+3 and k^2+k-3 are both prime.

Original entry on oeis.org

4, 7, 10, 22, 25, 34, 70, 79, 112, 130, 139, 172, 187, 217, 229, 262, 274, 295, 304, 322, 337, 364, 397, 400, 472, 499, 574, 580, 592, 622, 634, 655, 664, 697, 829, 844, 925, 1057, 1144, 1165, 1255, 1300, 1309, 1357, 1414, 1420, 1489, 1537, 1642, 1669, 1744

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 20 2009

			4 is in the list because 16+4+-3 = 23 and 17 are primes.
7 is in the list because 49+7+-3 = 53 and 59 are primes.

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

Cf. A088485, A027752

[k:k in [1..1750]| IsPrime(k^2+k+3) and IsPrime(k^2+k-3)]; // Marius A. Burtea, Feb 17 2020

q=3;lst3={};Do[p=n^2+n;If[PrimeQ[p-q]&&PrimeQ[p+q],AppendTo[lst3,n]],{n,0,7!}];lst3
Select[Range[2000],AllTrue[#^2+#+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 01 2019 *)

Definition rephrased by R. J. Mathar, Jun 27 2009

A161864 Numbers n such that n^2 + n + 5 and n^2 + n - 5 are both prime.

Original entry on oeis.org

3, 6, 11, 18, 21, 23, 27, 32, 42, 48, 51, 83, 86, 93, 116, 153, 158, 182, 188, 216, 282, 291, 317, 333, 396, 482, 681, 737, 786, 798, 818, 821, 872, 923, 956, 966, 977, 986, 1007, 1026, 1077, 1082, 1106, 1161, 1287, 1292, 1302, 1337, 1341, 1451, 1467, 1563

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 20 2009

			a(1)=3 as 12+-5 are primes. a(2)=6 as 42+-5 are primes.

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

Cf. A088485, A161863.

select(n -> isprime(n^2+n+5) and isprime(n^2+n-5), [$1..2000]); # Robert Israel, Nov 26 2017

q=5;lst5={};Do[p=n^2+n;If[PrimeQ[p-q]&&PrimeQ[p+q],AppendTo[lst5,n]], {n,0,7!}];lst5

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

A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 4, 4, 3, 5, 5, 4, 6, 6, 4, 4, 4, 3, 3, 4, 1, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 6, 4, 3, 5, 4, 4, 3, 5, 5, 6, 8, 6, 7, 11, 7, 6, 9, 8, 4, 8, 6, 5, 7, 5, 4, 8, 10, 5, 7, 9, 6, 10, 6, 7, 7, 7, 4, 4, 8, 5, 5, 4, 6, 9, 7, 7, 7, 7, 7, 8

Offset: 1

Zhi-Wei Sun, Oct 22 2013

nonn

Conjecture: a(n) > 0 for all n > 5.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
This implies that there are infinitely many twin prime pairs of the form {x^2 + x - 1, x^2 + x + 1}.
See also A230514 for a similar conjecture.

			a(8) = 1 since 8 = 2 + 3 + 3, and {2*3 - 1, 2*3 + 1} = {5, 7} and {3*4 - 1, 3*4 + 1} = {11, 13} are twin prime pairs.
a(39) = 1 since 39 = 3 + 15 + 21, and {3*4 - 1, 3*4 + 1} = {11, 13}, {15*16 - 1, 15*16 + 1} = {239, 241}, {21*22 - 1, 21*22 + 1} = {461, 463} are twin prime pairs.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A001359, A006512, A088485, A227923, A230219, A230252, A230351, A230514.

pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[n(n+1)+1]
a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

A236056 Numbers k such that k^2 +- k +- 1 is prime for all four possibilities.

Original entry on oeis.org

3, 6, 21, 456, 1365, 2205, 2451, 2730, 8541, 18486, 32199, 32319, 32781, 45864, 61215, 72555, 72561, 82146, 83259, 86604, 91371, 95199, 125334, 149331, 176889, 182910, 185535, 210846, 225666, 226254, 288420, 343161, 350091, 403941, 411501, 510399, 567204

Offset: 1

Derek Orr, Jan 18 2014

nonn

The only prime in this sequence is a(1) = 3.

			1365^2 + 1365 + 1 = 1864591,
1365^2 + 1365 - 1 = 1864589,
1365^2 - 1365 + 1 = 1861861, and
1365^2 - 1365 - 1 = 1861859 are all prime, so 1365 is a term of this sequence.

Numbers in the intersection of A002384, A045546, A055494, and A002328.

Numbers in the intersection of A131530 and A088485.

q:= k-> andmap(isprime, [seq(seq(k^2+i+j, j=[k, -k]), i=[1, -1])]):
select(q, [3*t$t=1..200000])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[568000],AllTrue[Flatten[{#^2+#+{1,-1},#^2-#+{1,-1}},1],PrimeQ]&] (* Harvey P. Dale, Jul 31 2022 *)

import sympy
from sympy import isprime
{print(p) for p in range(10**6) if isprime(p**2+p+1) and isprime(p**2-p+1) and isprime(p**2+p-1) and isprime(p**2-p-1)}

cp's OEIS Frontend

A088497 Duplicate of A088485.

Views

Author

Keywords

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A108309 Consider the triangle of odd numbers where the n-th row has the next n odd numbers. The sequence is the number of primes in the n-th row.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A088486 Primes p of the form k*(k + 1) - 1 such that p and p + 2 are twin primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A138303 Numbers k such that k*(k+1)-1 and k*(k+1)+1 are twin primes and k*(k+3)-1 and k*(k+3)+1 are also twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A161863 Numbers k such that k^2+k+3 and k^2+k-3 are both prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A161864 Numbers n such that n^2 + n + 5 and n^2 + n - 5 are both prime.

Views

Author

Keywords

Examples

Links

Crossrefs

A138303 Numbers k such that k(k+1)-1 and k(k+1)+1 are twin primes and k(k+3)-1 and k(k+3)+1 are also twin primes.