A056900 -id:A056900 - cp's OEIS frontend

A056899 Primes of the form k^2 + 2.

2, 3, 11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

Also, primes of the form k^2 - 2k + 3.
Note that all terms after the first two are equal to 11 modulo 72 and that (a(n)-11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime, and if k = 6*m+3 then a(n) = k^2 + 2 = 72*m*(m+1)/2 + 11.
The quotient cycle length is 2 in the continued fraction expansion of sqrt(p) for these primes. E.g.: cfrac(sqrt(6563),6) = 81+1/(81+1/(162+1/(81+1/(162+1/(81+1/(162+`...`)))))). - Labos Elemer, Feb 22 2001
Primes in A059100; except for a(2)=3 a subsequence of A007491 and congruent to 2 modulo 9. For n>2, a(n)=11 (mod 72). - M. F. Hasler, Apr 05 2009

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.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Intersection of A146327 and A000040; intersection of A059100 and A000040.

Cf. A002496.

[n: n in PrimesUpTo(175000) | IsSquare(n-2)];  // Bruno Berselli, Apr 05 2011

[ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // Vincenzo Librandi, Apr 06 2011

select(isprime, [seq(t^2+2, t = 0..1000)]); # Robert Israel, Sep 03 2015

Select[ Range[0, 500]^2 + 2, PrimeQ] (* Robert G. Wilson v, Sep 03 2015 *)

print1("2, 3");forstep(n=3,1e4,6,if(isprime(t=n^2+2),print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011

For n>1, a(n) = 72*A000217(A056900(n-2))+11

a(n) = A067201(n)^2 + 2. - M. F. Hasler, Apr 05 2009

A067201 Numbers k such that k^2 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, 105, 111, 117, 123, 147, 171, 219, 225, 237, 243, 249, 255, 273, 297, 303, 309, 321, 345, 351, 363, 369, 375, 387, 417, 423, 429, 441, 447, 453, 477, 501, 513, 549, 555, 561, 573, 603, 609, 651, 675, 681, 699, 711, 753

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

All terms > 1 are divisible by 3. - Robert Israel, Sep 05 2014

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Equals 6*A056900(n-2) + 3, n>1.
Cf. A106571, A040976.
Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), this sequence (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

select(t -> isprime(t^2+2), [0,1,seq(3*i,i=1..1000)]); # Robert Israel, Sep 05 2014

lst={};Do[If[PrimeQ[n^2+2], AppendTo[lst, n]], {n, 3*10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Join[{0, 1}, Select[Range[3, 1000, 6], PrimeQ[#^2 + 2] &]] (* Zak Seidov, Jan 30 2014 *)

select(n -> isprime(n^2+2),[1..500]) \\ Edward Jiang, Sep 05 2014

A056906 Numbers k such that 36*k^2 + 5 is prime.

Original entry on oeis.org

0, 1, 2, 6, 8, 12, 13, 16, 19, 21, 27, 28, 33, 34, 41, 43, 49, 56, 57, 62, 69, 72, 76, 77, 82, 84, 86, 89, 92, 96, 98, 99, 104, 111, 119, 121, 126, 128, 131, 132, 133, 134, 139, 142, 146, 148, 153, 159, 166, 168, 169, 173, 174

Offset: 1

Henry Bottomley, Jul 07 2000

Except for a(1), a(n) is never a multiple of 5.

			a(3)=2 since 36*2^2 + 5 = 149, which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

This sequence and formula generate all primes of the form k^2+5, i.e., A056905.
Except for the first term, this sequence is a subsequence of A047201.
Cf. A056900, A056902.

[n: n in [0..200]| IsPrime(36*n^2+5)]; // Vincenzo Librandi, Jul 14 2012

Select[Range[0,200],PrimeQ[36#^2+5]&] (* Harvey P. Dale, Jul 25 2011 *)

is(n)=isprime(36*n^2+5) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = sqrt(A056905(n)-5)/6.

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A091199 Numbers k such that (6k-3)^2 + 2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 14, 17, 18, 19, 20, 21, 25, 29, 37, 38, 40, 41, 42, 43, 46, 50, 51, 52, 54, 58, 59, 61, 62, 63, 65, 70, 71, 72, 74, 75, 76, 80, 84, 86, 92, 93, 94, 96, 101, 102, 109, 113, 114, 117, 119, 126, 130, 135, 137, 140, 145, 148, 150, 151, 152, 156, 160

Offset: 1

Zak Seidov, Feb 22 2004

Arises from A056899: primes of the form m^2+2; m should be of the form 6n-3, hence this sequence.

			10 is a member because (60-3)^2 + 2 = 3251 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A056899, A056900.

[n: n in [1..250]| IsPrime((6*n-3)^2+2)]; // Vincenzo Librandi, Jul 16 2012

Select[ Range[ 163], PrimeQ[(6# - 3)^2 + 2] &] (* Robert G. Wilson v, Feb 26 2004 *)
lst={};Do[If[PrimeQ[(6n-3)^2+2],AppendTo[lst,n]],{n,500}];lst (* Vincenzo Librandi, Jul 16 2012 *)

is(n)=isprime((6*n-3)^2+2) \\ Charles R Greathouse IV, May 22 2017

a(n) = A056900(n-1) + 1. - Jeppe Stig Nielsen, May 14 2017

More terms from Ray Chandler and Robert G. Wilson v, Feb 25 2004

A056908 Numbers k such that 36k^2 + 36k + 13 is prime.

Original entry on oeis.org

0, 2, 4, 5, 7, 9, 14, 19, 22, 24, 29, 30, 34, 40, 42, 44, 50, 59, 62, 70, 72, 74, 75, 79, 80, 82, 84, 95, 102, 110, 119, 125, 132, 135, 139, 149, 150, 157, 160, 165, 172, 180, 197, 199, 200, 209, 210, 212, 224, 225, 227, 229, 230, 232, 235, 240, 244, 249

Offset: 1

Henry Bottomley, Jul 07 2000

36*k^2 + 36*k + 13 = (6*k+3)^2 + 4, which is 4 more than a square.

			a(2)=4 since 36*4^2 + 36*4 + 13 = 733, which is prime (as well as being four more than a square).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

This sequence and formula, together with A056907 and its formula, generate all primes of the form k^2+4, i.e., A005473.

Cf. A056900, A056902, A056904, A056906.

[n: n in [0..70]| IsPrime(36*n^2+36*n+13)]; // Vincenzo Librandi, Jul 14 2012

Select[Range[0,700],PrimeQ[36#^2+36#+13]&] (* Vincenzo Librandi, Jul 14 2012 *)

is(n)=isprime(36*n^2+36*n+13) \\ Charles R Greathouse IV, Mar 01 2017

A056907 Numbers k such that 36k^2 + 12k + 5 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, 1, 2, -3, -6, 6, -8, -11, 11, 12, 14, -16, 16, 17, 19, -21, -23, -26, 27, -28, 32, -34, -36, 36, -39, 39, -41, 42, 44, -46, 46, -48, -49, 51, 52, -53, -58, 62, 64, 67, -68, -71, 71, -76, 77, 79, 81, -84, -89, 91, 96, -99, -101, 101, 102, -104, -111, 111, -113

Offset: 0

Henry Bottomley, Jul 07 2000

sign

36*k^2 + 12*k + 5 = (6*k+1)^2 + 4, which is four more than a square. Except for a(0), a(n) is never a multiple of 5.

			a(3)=2 since 36*2^2 + 12*2 + 5 = 173 which is prime (as well as being four more than a square).

This sequence and formula, together with A056908 and its formula, generate all primes of the form k^2+4, i.e., A005473. Except for the first term, this sequence is a subsequence of A047201. Cf. A056900, A056902, A056904, A056906.

A056910 Numbers k such that 36k^2 + 12k + 7 is prime (sorted by absolute values with negatives before positives).

Original entry on oeis.org

0, -1, -2, 3, 4, 5, -6, 10, -11, 13, -15, 15, 18, -22, 24, 25, 29, -31, 33, -37, -45, -55, 55, 59, -67, -72, 74, 80, -81, 85, -86, 88, -90, -95, 99, -101, -102, 108, -116, 118, -122, 129, -130, 143, 148, -151, -155, -157, 158, 159, -162, 164, 165

Offset: 0

Henry Bottomley, Jul 07 2000

sign

36*k^2 + 12*k + 7 = (6*k+1)^2 + 6, which is six more than a square.

			a(2)=-2 since 36*(-2)^2 + 12*(-2) + 7 = 127, which is prime (as well as being six more than a square).

This sequence and formula generate all primes of the form k^2+6, i.e., A056909. Except for the first term, none of the a(n) are a multiple of 7 and so the rest of this sequence is a subsequence of A047304. Cf. A056900, A056902, A056904, A056906, A056907, A056908.

a(n) = (-1 +- sqrt(A056909(n) - 6))/6, choosing +- to give an integer result for each n.

A334294 Numbers k such that 70k^2 + 70k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 46, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 74, 76, 77, 78, 79, 80, 81, 82, 87, 88, 90, 93, 96, 97, 100

Offset: 1

James R. Buddenhagen, Apr 21 2020

nonn

Among quadratic polynomials in k of the form a*k^2 + a*k - 1 the value a=70 gives the most primes for any a in the range 1<=a<=300, at least up to k=40000. Here a and k are positive integers. Other "good" values of a are a=250, a=99, and a=19.

			For k=1, 70*k^2 + 70*k - 1 = 70*1^2 + 70*1 - 1 = 139, which is prime, so 1 is in the sequence.

N. Boston et M. L.Greenwood, Quadratics representing primes, Amer. Math. Monthly 102:7 (1995), 595-599.
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A001912, A002384, A005574, A027861, A028870, A056900.

Cf. A067201, A088572, A089623, A091199, A271980.

a:=proc(n) if isprime(70*n^2+70*n-1) then n else NULL end if end proc;
seq(a(n),n=1..100);

Select[Range@ 100, PrimeQ[70 #^2 + 70 # - 1] &] (* Michael De Vlieger, May 26 2020 *)

cp's OEIS Frontend

A056899 Primes of the form k^2 + 2.

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A067201 Numbers k such that k^2 + 2 is prime.

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A056906 Numbers k such that 36*k^2 + 5 is prime.

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A091199 Numbers k such that (6k-3)^2 + 2 is prime.

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A056908 Numbers k such that 36*k^2 + 36*k + 13 is prime.

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A056907 Numbers k such that 36*k^2 + 12*k + 5 is prime (sorted by absolute values with negatives before positives).

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A056910 Numbers k such that 36*k^2 + 12*k + 7 is prime (sorted by absolute values with negatives before positives).

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A056908 Numbers k such that 36k^2 + 36k + 13 is prime.

A056907 Numbers k such that 36k^2 + 12k + 5 is prime (sorted by absolute values with negatives before positives).

A056910 Numbers k such that 36k^2 + 12k + 7 is prime (sorted by absolute values with negatives before positives).

A334294 Numbers k such that 70k^2 + 70k - 1 is prime.