A092968 -id:A092968 - cp's OEIS frontend

A050265 Primes of the form 2*n^2 + 11.

11, 13, 19, 29, 43, 61, 83, 109, 139, 173, 211, 349, 461, 523, 659, 733, 811, 1069, 1163, 1579, 1693, 1811, 1933, 2749, 3373, 3539, 3709, 4243, 4813, 5011, 5419, 5843, 7211, 7699, 7949, 8461, 9533, 9811, 10093, 11261, 13789, 14461, 15149, 16573

Offset: 1

Eric W. Weisstein

The polynomial 2*n^2 + 11 first fails to produce a prime for n = 11, giving 253 = 11 * 23. - Alonso del Arte, Sep 04 2016

Let P(n) = 2*n^2 + 11. The polynomial P(2*n - 10) = 8*n^2 - 80*n + 11 produces prime values (not distinct) for n = 0 to 10. The polynomial P(3*n - 19) = 18*n^2 - 228*n + 733 produces distinct prime values for n = 0 to 9. - Peter Bala, Apr 16 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A005846, A007641.

[a: n in [0..100] | IsPrime(a) where a is 2*n^2+11]; // Vincenzo Librandi, Dec 08 2011

Select[2Range[0, 100]^2 + 11, PrimeQ] (* Harvey P. Dale, May 20 2011 *)

a(n) = 11 + 2*(A092968(n))^2. - R. J. Mathar, Jan 03 2009

11 added by Vincenzo Librandi, Dec 08 2011

A092969 a(1) = 2; for n>1, a(n) = largest prime of the form n!/k + 1, where k < n, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 7, 13, 61, 241, 2521, 20161, 72577, 604801, 39916801, 59875201, 3113510401, 17435658241, 186810624001, 10461394944001, 118562476032001, 0, 24329020081766401, 304112751022080001, 12772735542927360001

Offset: 1

Amarnath Murthy, Mar 26 2004

nonn

Conjecture: There are only finitely many zeros in this sequence. In other words the sequence is identical to A092965 barring a finite set of terms which are zero.

I found zeros for n: 18,51,53,84,95,100,104,106,143,178,180,181,188,202,203,(204). - Robert G. Wilson v, Mar 27 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A092968, A092970.

f[n_] := Block[{k = 1}, While[ !PrimeQ[n!/k + 1], k++ ]; If[k < n, n!/k + 1, 0]]; Table[ f[n], {n, 22}] (* Robert G. Wilson v, Mar 27 2004 *)

a(n)=for (i=1,n,if(isprime(n!/i+1),return((n!/i+1))))

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

A092970 Smallest prime of the form n!/k + 1. k < = n, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 3, 7, 31, 181, 1009, 13441, 45361, 453601, 3991681, 39916801, 566092801, 10897286401, 130767436801, 2988969984001, 25406244864001, 0, 8109673360588801, 304112751022080001, 2688996956405760001

Offset: 1

Amarnath Murthy, Mar 26 2004

nonn

			a(10) = 453601 = 10!/8 + 1, as 10!/10 + 1 and 10!/9 + 1 are both composite.

Cf. A092968, A092969, A092068.

Table[SelectFirst[Reverse[n!/Range[n]+1],PrimeQ],{n,30}]/.(Missing[ "NotFound"] -> 0) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2019 *)

a(n)=for (i=1,n,if(isprime(n!/(n-i+1)+1),return((n!/(n-i+1)+1))))

Corrected and extended by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

A260352 Numbers n such that both 2n^2+11 and 2(n+1)^2+11 are prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 18, 19, 23, 28, 29, 30, 41, 42, 49, 62, 69, 70, 94, 95, 108, 123, 136, 145, 151, 152, 189, 190, 204, 212, 215, 223, 227, 276, 277, 278, 281, 290, 291, 294, 314, 328, 342, 353, 367, 368, 372, 405, 410, 436, 488, 497

Offset: 1

Zak Seidov, Jul 23 2015

Both n and n+1 are terms in A092968.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A092968.

[n: n in [0..600]| IsPrime( 2*n^2+11) and IsPrime(2*(n+1)^2+11)]; // Vincenzo Librandi, Jul 26 2015

Select[Range[0, 600], PrimeQ[2 #^2 + 11] &&  PrimeQ[2 (# + 1)^2 + 11] &] (* Vincenzo Librandi, Jul 26 2015 *)

isok(n) = isprime(2*n^2+11) && isprime(2*(n+1)^2+11); \\ Michel Marcus, Jul 26 2015

A255842 a(n) = 2*n^2 + 12.

Original entry on oeis.org

12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
Equivalently, numbers m such that 2*m - 24 is a square.
For n = 0..10, a(n) - 1 is prime (see A092968).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016825 (first differences), A092968, A114949.
Subsequence of A047238 and A047406.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+12: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 12, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+12)
```
Sage
```
[2*n^2+12 for n in (0..50)]
```

G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A114949(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
E.g.f.: 2*exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 24 2025

Edited by Bruno Berselli, Mar 11 2015

A257648 Numbers m such that both p=2m^2+11 and q=2p^2+11 are prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 13, 20, 31, 52, 54, 62, 65, 70, 75, 137, 151, 153, 163, 212, 224, 281, 284, 329, 384, 419, 424, 445, 455, 489, 505, 524, 581, 593, 642, 646, 680, 706, 723, 738, 746, 775, 787, 795, 830, 841, 843, 918, 953, 970, 973, 984

Offset: 1

Zak Seidov, Jul 25 2015

nonn

Numbers m such that both m and p=2*m^2+11 are terms in A092968. Also, both p and q are terms in A050265.

			n=1: m=1=A092968(2) and p=13=A092968(12),
n=8: m=13=A092968(12) and p=349=A092968(127),
n=9: m=20=A092968(12) and p=811=A092968(251).

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

Cf. A050265, A092968.

Reap[Do[If[PrimeQ[p=2*k^2+11]&&PrimeQ[2*(p)^2+11],Sow[k]],{k,10^3}]][[2,1]]
bprQ[n_]:=Module[{p=2n^2+11},AllTrue[{p,2p^2+11},PrimeQ]]; Select[Range[ 1000],bprQ] (* Harvey P. Dale, Jun 16 2022 *)

A260354 Numbers n such that 2n^2+11, 2(n+1)^2+11 and 2*(n+2)^2+11 are prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 18, 28, 29, 41, 69, 94, 151, 189, 276, 277, 290, 367, 497, 578, 579, 580, 617, 618, 619, 620, 744, 810, 887, 903, 939, 1048, 1049, 1108, 1124, 1125, 1172, 1303, 1304, 1305, 1399, 1420, 1449, 1614, 1761, 1790, 1838, 1861, 1865, 1971

Offset: 1

Zak Seidov, Jul 23 2015

n, n+1 and n+2 are terms in A092968, i.e., n and n+1 are terms in A260352.

Zak Seidov, Table of n, a(n) for n = 1..10000

Subsequence of A260352 and A092968.

[n: n in [0..3000]| IsPrime( 2*n^2+11) and IsPrime(2*(n+1)^2+11) and IsPrime(2*(n+2)^2+11)]; // Vincenzo Librandi, Jul 26 2015

Select[Range[0, 2000], PrimeQ[2 #^2 + 11] && PrimeQ[2 (# + 1)^2 + 11] && PrimeQ[2 (# + 2)^2 + 11] &] (* Vincenzo Librandi, Jul 26 2015 *)

A174131 Either 2*n^2-+11 is a prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 12, 13, 16, 18, 19, 20, 23, 24, 27, 28, 29, 31, 37, 41, 43, 46, 48, 49, 50, 52, 60, 62, 65, 70, 71, 72, 78, 83, 85, 87, 90, 91, 94, 95, 96, 98, 100, 105, 106, 108, 109, 111, 117, 120, 123, 124, 126, 128, 130, 134, 136, 137, 139, 145, 146, 147, 151

Offset: 1

Juri-Stepan Gerasimov, Mar 09 2010

nonn

			a(1)=0 because 2*0^2-11=-11=nonprime and 2*0^2+11=11=prime; a(2)=1 because 2*1^2-11=-9=nonprime and 2*1^2+11=13=prime; a(3)=2 because 2*2^2-11=-3=nonprime and 2*2^2+11=19=prime; a(4)=4 because 2*4^2-11=21=nonprime and 2*4^2+11=43=prime.

Cf. A050265, A092968.

Corrected (15 removed) and extended beyond 71 by R. J. Mathar, Apr 20 2010

A260826 Let f(k)=2*k^2+11. For n=0,1,...,11, a(n) = smallest m >= 0 such that f(m-1) is composite if m>0, f(m), f(m+1), ...,f(m+n-1) are prime, and f(m+n) is composite.

Original entry on oeis.org

11, 13, 15, 18, 28, 578, 617, 2067795, 843755046, 134239787815, 1434279786435, 0

Offset: 0

Giovanni Resta and Zak Seidov, Aug 01 2015

Sequence is complete, a(9) and a(10) are due to Giovanni Resta.

			Note that f(0), f(1), ..., f(10) are prime, while f(11) and f(12) are composite. So a(0)=11 and a(11)=0.

Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A050265, A092968.

cp's OEIS Frontend

A050265 Primes of the form 2*n^2 + 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A092969 a(1) = 2; for n>1, a(n) = largest prime of the form n!/k + 1, where k < n, or 0 if no such prime exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A092970 Smallest prime of the form n!/k + 1. k < = n, or 0 if no such prime exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A260352 Numbers n such that both 2*n^2+11 and 2*(n+1)^2+11 are prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A255842 a(n) = 2*n^2 + 12.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A257648 Numbers m such that both p=2*m^2+11 and q=2*p^2+11 are prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A260354 Numbers n such that 2*n^2+11, 2*(n+1)^2+11 and 2*(n+2)^2+11 are prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A174131 Either 2*n^2-+11 is a prime.

A260352 Numbers n such that both 2n^2+11 and 2(n+1)^2+11 are prime.

A257648 Numbers m such that both p=2m^2+11 and q=2p^2+11 are prime.

A260354 Numbers n such that 2n^2+11, 2(n+1)^2+11 and 2*(n+2)^2+11 are prime.