A066049 -id:A066049 - cp's OEIS frontend

A090697 Numbers n such that n^2/2 - 1 is a prime.

4, 6, 8, 12, 14, 16, 20, 22, 26, 30, 34, 36, 42, 44, 48, 50, 56, 68, 72, 76, 78, 82, 84, 86, 90, 92, 98, 100, 104, 112, 118, 124, 126, 128, 138, 146, 152, 160, 162, 170, 174, 182, 184, 190, 196, 204, 216, 218, 224, 226, 230, 236, 250, 252, 254, 264, 268, 274, 280

Offset: 1

Giovanni Teofilatto, Dec 20 2003

A066436 gives resulting primes p such that 2p+2 is square. - Ray Chandler, Dec 25 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

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

Cf. A066049, A066436, A091176, A110558.

Select[Range[2,300,2],PrimeQ[#^2/2-1]&] (* Harvey P. Dale, Apr 05 2014 *)

isok(n) = !(n % 2) && isprime(n^2/2 - 1); \\ Michel Marcus, Jul 23 2016

a(n) = 2*A066049(n) = A110558(n)/2. - Ray Chandler, Dec 25 2003

Corrected and extended by Ray Chandler, Dec 25 2003

A110558 Numbers n such that (n^2-8)/8 is prime.

Original entry on oeis.org

8, 12, 16, 24, 28, 32, 40, 44, 52, 60, 68, 72, 84, 88, 96, 100, 112, 136, 144, 152, 156, 164, 168, 172, 180, 184, 196, 200, 208, 224, 236, 248, 252, 256, 276, 292, 304, 320, 324, 340, 348, 364, 368, 380, 392, 408, 432, 436, 448, 452, 460, 472, 500, 504, 508

Offset: 1

Pierre CAMI, Sep 12 2005

nonn

These numbers need to be of the form 4*j then (16*j^2-8)/8 = 2*j^2-1.

A066436 gives resulting primes p such that 8p+8 is square. - Ray Chandler, Sep 15 2005

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A066049, A066436, A090697, A091176.

Select[Range[510],PrimeQ[(#^2-8)/8]&] (* Harvey P. Dale, Feb 03 2011 *)

is(n)=isprime((n^2-8)/8) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = 2*A090697(n) = 4*A066049(n). - Ray Chandler, Sep 15 2005

Extended by Ray Chandler, Sep 15 2005

A143830 Primes of the form 12*n^2-1.

Original entry on oeis.org

11, 47, 107, 191, 431, 587, 971, 1451, 2027, 2351, 2699, 3467, 4799, 5807, 6911, 7499, 8111, 8747, 10091, 10799, 14699, 15551, 16427, 17327, 18251, 25391, 27647, 36299, 41771, 44651, 55487, 57131, 62207, 67499, 71147, 74891, 80687, 92927, 99371

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Equals A089682 without the 2. [Sketch of proof: the primes 3*n^2-1 are odd if 2 is left out, so 3*n^2 is even, so n^2 is even, so n is even = 2*k. 3*(2*k)^2-1 = 12*k^2-1.] [From R. J. Mathar, Sep 04 2008]

A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829

p = 12; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 1000}]; a

A143835 a(n) = Number of x <= 10^n such that 2x^2-1 is prime.

Original entry on oeis.org

7, 45, 303, 2202, 17185, 141444, 1200975, 10448345, 92435171, 828797351, 7511268020, 68680339342

Offset: 1

Artur Jasinski, Sep 02 2008, Sep 04 2008

nonn

			a(1) = 7 because are 7 different x ={2, 3, 4, 6, 7, 8, 10} <= 10^1 where 2x^2-1 is prime = {7, 17, 31, 71, 97, 127, 199}.

Bernhard Helmes, Prime sieving on the polynomial f(n)=2n^2-1.

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143832, A143833, A143834.

l = 0; p = 2; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], l = l + 1]; If[N[Log[x]/Log[10]] == Round[N[Log[x]/Log[10]]], Print[l]; AppendTo[a, l]], {x, 1, 10000000}]; a (*Artur Jasinski*)

Added link and extended to agree with website. - Ray Chandler, Jun 30 2015

A143831 Numbers n such that 12n^2 - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 15, 17, 20, 22, 24, 25, 26, 27, 29, 30, 35, 36, 37, 38, 39, 46, 48, 55, 59, 61, 68, 69, 72, 75, 77, 79, 82, 88, 91, 93, 94, 102, 105, 107, 108, 115, 116, 117, 118, 121, 124, 130, 134, 136, 137, 140, 149, 152, 154, 157, 158, 159, 162, 167

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830.

p = 12; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a

is(n)=isprime(12*n^2-1) \\ Charles R Greathouse IV, Feb 20 2017

A143833 Numbers n such that 14n^2 - 1 is prime.

Original entry on oeis.org

1, 4, 5, 6, 10, 11, 16, 21, 26, 34, 36, 44, 45, 49, 54, 55, 59, 65, 69, 71, 76, 80, 85, 91, 95, 96, 100, 104, 106, 110, 114, 115, 120, 121, 125, 135, 139, 166, 169, 176, 180, 190, 195, 201, 204, 206, 214, 226, 230, 231, 234, 241, 254, 256, 264, 265, 269, 270, 275, 280

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

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

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143832.

p = 14; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[300],PrimeQ[14#^2-1]&] (* Harvey P. Dale, Aug 29 2011 *)

is(n)=isprime(14*n^2-1) \\ Charles R Greathouse IV, Feb 20 2017

A240235 Least number k such that k*n^k - 1 is prime. a(n) = 0 if no such number exists.

Original entry on oeis.org

3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1

Offset: 1

Derek Orr, Apr 02 2014

nonn

a(n) = 1 iff n-1 is prime.

a(145) is either 0 or > 275000. - Robert G. Wilson v, Jan 23 2017

			1*1^1 - 1 = 0 is not prime. 2*1^2 - 1 = 1 is not prime. 3*1^3 - 1 = 2 is prime. Thus, a(1) = 3.

Robert G. Wilson v, Table of n, a(n) for n = 1..144
Steven Harvey, Generalized Woodall Primes

Cf. A066049, A239787, A281141.

a(n)=k=1;while(!ispseudoprime(k*n^k-1),k++);return(k); n=1;while(n<100,print(a(n));n+=1)

a(23) and a(29) given using link. - Derek Orr, Aug 16 2014

A119591 Least k such that 2*n^k - 1 is prime.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 1, 10, 1, 1, 6, 1, 2, 6, 1, 2, 136, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 2, 4, 4, 1, 2, 1, 1, 44, 1, 1, 2, 1, 3, 2, 5, 3, 2, 2, 1, 4, 1, 768, 4, 1, 1, 52, 34, 2, 132, 1, 1, 14, 7, 1, 2, 2, 1, 8, 1, 2, 10, 1, 24, 60, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1

Offset: 2

Pierre CAMI, Jun 01 2006

From Eric Chen, Jun 01 2015: (Start)
Conjecture: a(n) is defined for all n.
a(303) > 10000, a(304)..a(360) = {1, 2, 11, 1, 990, 1, 1, 2, 2, 4, 74, 5, 1, 10, 6, 6, 4, 1, 1, 2, 1, 9, 12, 1, 80, 2, 1, 1, 2, 14, 3, 2, 3, 1, 12, 1, 60, 36, 1, 8, 4, 34, 1, 522, 3, 15, 14, 1, 6, 2, 3, 1, 4, 5, 4, 10, 1}.
a(n) = 1 if and only if n is in A006254. (End)
From Eric Chen, Sep 16 2021: (Start)
Now a(303) is known to be 40174, also other terms > 10000: a(383) = 20956, a(515) = 58466, a(522) = 62288, a(578) = 129468, a(581) > 400000, a(590) = 15526, a(647) = 21576, a(662) = 16590, a(698) = 127558, a(704) = 62034, see the a-file and the references.
a(n) = 2 if and only if n is in A066049 but not in A006254.
a(n) = 3 if and only if n is in A214289 but not in A006254 or A066049. (End)

Eric Chen, Table of n, a(n) for n = 2..580
Gary Barnes, Riesel conjectures and proofs
Eric Chen, Table of n, a(n) for n = 2..2050 status
Prime Wiki, Riesel prime small bases least n

Cf. A119624, A253178.

Numbers r such that 2*k^r-1 is prime: A090748 (k=2), A003307 (k=3), A146768 (k=4), A120375 (k=5), A057472 (k=6), A002959 (k=7), ... (k=8), ... (k=9), A002957 (k=10), A120378 (k=11), ... (k=12), A174153 (k=13), A273517 (k=14), ... (k=15), ... (k=16), A193177 (k=17), A002958 (k=25).

f[n_] := Block[{k = 0}, While[ ! PrimeQ[2*n^k - 1], k++ ]; k ]; Table[f[n], {n, 2, 106}] (* Ray Chandler, Jun 08 2006 *)

a(n) = for(k=1, 2^24, if(ispseudoprime(2*n^k-1), return(k))) \\ Eric Chen, Jun 01 2015

From Eric Chen, Sep 16 2021: (Start)
a(6*n) = A098873(n).
a(2^n) = A279095(n).
a(A006254(n)) = 1.
a(A066049(n)) <= 2.
a(A214289(n)) <= 3. (End)

Corrected and extended by Ray Chandler, Jun 08 2006

A143832 Primes of the form 14 n^2-1.

Original entry on oeis.org

13, 223, 349, 503, 1399, 1693, 3583, 6173, 9463, 16183, 18143, 27103, 28349, 33613, 40823, 42349, 48733, 59149, 66653, 70573, 80863, 89599, 101149, 115933, 126349, 129023, 139999, 151423, 157303, 169399, 181943, 185149, 201599, 204973, 218749

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Primes of the form k n^2-1 k = 2 A066436 these n are A066049 k = 4 only one prime 3 when n = 1 k = 6 A090686 these n are A143826 k = 8 A090684 these n are A143827 k =10 A143828 these n are A143829 k =12 A143830 these n are A143831 k =14 A143832 these n are A143833 k =16 lack of primes

A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143833

p = 14; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 1000}]; a
Select[14*Range[200]^2-1,PrimeQ] (* Harvey P. Dale, Jul 29 2024 *)

A160697 Record values in A160696.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 18, 21, 22, 24, 25, 28, 34, 36, 38, 39, 41, 42, 43, 45, 46, 49, 50, 52, 56, 59, 62, 63, 64, 69, 73, 76, 80, 81, 85, 87, 91, 92, 95, 98, 102, 108, 109, 112, 113, 115, 118, 125, 126, 127, 132, 134, 137, 140, 141, 143, 153, 154, 155

Offset: 1

Reinhard Zumkeller, May 24 2009

nonn

a(n)=A160696(A160698(n)) and A160696(m)A160698(n);

for n>1: a(n)=A066049(n-1) and A066436(n-1)+1=2*a(n)^2.

cp's OEIS Frontend

A090697 Numbers n such that n^2/2 - 1 is a prime.

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A110558 Numbers n such that (n^2-8)/8 is prime.

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A143830 Primes of the form 12*n^2-1.

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A143835 a(n) = Number of x <= 10^n such that 2x^2-1 is prime.

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A143831 Numbers n such that 12n^2 - 1 is prime.

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A143833 Numbers n such that 14n^2 - 1 is prime.

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A240235 Least number k such that k*n^k - 1 is prime. a(n) = 0 if no such number exists.

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A119591 Least k such that 2*n^k - 1 is prime.

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