A005574 -id:A005574 - cp's OEIS frontend

A114271 Numbers k such that k^2 + 8 is prime.

3, 9, 15, 21, 33, 51, 57, 81, 87, 111, 117, 123, 129, 135, 141, 147, 153, 177, 189, 213, 219, 255, 279, 285, 315, 321, 327, 345, 351, 363, 399, 417, 465, 471, 477, 483, 495, 549, 579, 585, 627, 657, 663, 669, 723, 735, 741, 747, 759, 771, 783, 789, 807, 825

Offset: 1

Zak Seidov, Nov 19 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), this sequence (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

a={};Do[If[PrimeQ[n^2+8], AppendTo[a, n]], {n,0,10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Range[1,901,2],PrimeQ[#^2+8]&] (* Harvey P. Dale, Aug 27 2023 *)

is(n)=isprime(n^2+8) \\ Charles R Greathouse IV, Jan 21 2015

A103854 Positive integers n such that n^6 + 1 is semiprime.

Original entry on oeis.org

2, 4, 10, 36, 56, 94, 126, 224, 260, 270, 300, 350, 686, 716, 780, 1036, 1070, 1080, 1156, 1174, 1210, 1394, 1416, 1434, 1440, 1460, 1524, 1550, 1576, 1616, 1654, 1660, 1700, 1756, 1860, 1980, 2054, 2084, 2096, 2116, 2224, 2454, 2600, 2664, 2770, 2864

Offset: 1

Jonathan Vos Post, Mar 31 2005

n^6+1 can only be prime when n = 1, n^6+1 = 2. This is because the sum of cubes formula gives the polynomial factorization n^6+1 = (n^2+1) * (n^4 - n^2 + 1). Hence n^6+1 can only be semiprime when both (n^2+1) and (n^4 - n^2 + 1) are primes.

			n n^6+1 = (n^2+1) * (n^4 - n^2 + 1)
2 65 = 5 * 13
4 4097 = 17 * 241
10 1000001 = 101 * 9901
36 2176782337 = 1297 * 1678321
56 30840979457 = 3137 * 9831361
94 689869781057 = 8837 * 78066061
126 4001504141377 = 15877 * 252031501
224 126324651851777 = 50177 * 2517580801

Robert Price, Table of n, a(n) for n = 1..1134

Subsequence of A005574.

Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ 2Range@1526, semiprimeQ[ #^6 + 1] &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[200000], PrimeQ[#^2 + 1] && PrimeQ[(#^6 + 1)/(#^2 + 1)] &] (* Robert Price, Mar 11 2015 *)

is(n)=my(s=n^2); isprime(s+1) && isprime(s^2-s+1) \\ Charles R Greathouse IV, Aug 31 2021

a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.

More terms from Robert G. Wilson v, May 26 2006

A096012 Numbers k such that k^2+1 and (k+2)^2+1 are both prime; twin k^2+1 primes.

Original entry on oeis.org

2, 4, 14, 24, 54, 124, 204, 384, 464, 634, 644, 714, 1094, 1144, 1174, 1244, 1274, 1314, 1374, 1564, 1614, 1674, 1684, 1964, 2054, 2084, 2094, 2404, 2454, 2534, 2664, 2834, 2924, 3134, 3304, 3534, 3754, 3774, 4024, 4154, 4174, 4364, 4604, 4614, 4734, 4784

Offset: 1

Jason Earls, Jul 20 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Cf. A005574, A108814.

[n: n in [1..5000] | IsPrime(n^2+1) and IsPrime((n+2)^2+1)]; // Vincenzo Librandi, Feb 27 2016

Select[Range[5000],AllTrue[{#^2+1,(#+2)^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 23 2014 *)
Select[Range[5000], PrimeQ[#^2 + 1] && PrimeQ[(# + 2)^2 + 1] &] (* Vincenzo Librandi, Feb 27 2016 *)

isok(n) = isprime(n^2+1) && isprime((n+2)^2+1); \\ Michel Marcus, Feb 27 2016

a(k) = A108814(k) - 1. - Jeppe Stig Nielsen, Feb 26 2016

A113536 Numbers k such that k^2 + 13 is prime.

Original entry on oeis.org

0, 2, 4, 10, 12, 16, 18, 28, 40, 42, 44, 46, 60, 68, 72, 82, 84, 88, 94, 108, 110, 114, 116, 122, 126, 142, 144, 152, 158, 180, 192, 194, 198, 200, 220, 222, 264, 266, 268, 282, 284, 296, 298, 332, 336, 340, 354, 378, 380, 418, 420, 430, 434, 446, 464, 466, 486

Offset: 1

Zak Seidov, Jan 13 2006

nonn

			If n=40 then n^2 + 13 = 1613 (prime), so 40 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other cases: A005574 k=1, A067201 k=2, A049422 k=3, A007591 k=4, A078402 k=5, A114269-A114275 k=6-12.

With[{k=13}, Select[Range[1000], PrimeQ[ #^2+k]&]]

is(n)=isprime(n^2+13) \\ Charles R Greathouse IV, Feb 17 2017

Edited by R. J. Mathar, Aug 07 2008

A125260 Numbers k such that k^5 + 4 is prime.

Original entry on oeis.org

1, 7, 9, 25, 39, 45, 73, 85, 99, 147, 163, 165, 169, 189, 213, 219, 223, 225, 249, 253, 259, 279, 333, 337, 385, 433, 457, 465, 469, 477, 499, 595, 639, 643, 655, 703, 709, 715, 729, 849, 853, 895, 969, 973, 979, 987, 1065, 1075, 1077, 1093, 1165, 1239, 1255

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..4431

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), this sequence (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)...

lst={};k=5;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[1300],PrimeQ[#^5+4]&] (* Harvey P. Dale, Oct 14 2019 *)

is(n)=isprime(n^5+4) \\ Charles R Greathouse IV, Feb 17 2017

A035092 Smallest k such that (n^2)*k + 1 is prime.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 3, 4, 1, 8, 1, 12, 4, 30, 1, 2, 3, 24, 1, 18, 1, 2, 4, 12, 2, 16, 12, 2, 3, 6, 1, 4, 13, 6, 1, 10, 2, 12, 6, 2, 6, 4, 8, 6, 9, 6, 9, 28, 1, 4, 1, 10, 3, 6, 4, 46, 4, 4, 3, 4, 1, 4, 3, 22, 6, 10, 2, 4, 1, 2, 7, 22, 3, 6, 4, 6, 3, 10, 1, 4, 3, 2, 4, 6, 1, 10, 4, 2, 1

Offset: 1

Labos Elemer

			a(40) = 1 because in 1600k + 1 at k = 1, 1601 is the smallest prime;
a(61) = 46 because in the 46*46*k + 1 sequence the first prime appears at k = 46; it is 171167.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. See also A005574 and A002496.

Table[k = 1; While[! PrimeQ[k (n^2) + 1], k++]; k, {n, 94}] (* Michael De Vlieger, Dec 17 2016 *)

a(n)=k=1;while(!isprime(k*n^2+1),k++);k
vector(100,n,a(n)) \\ Derek Orr, Oct 01 2014

A125259 Numbers k such that k^4 + 3 is prime.

Original entry on oeis.org

0, 2, 8, 16, 22, 26, 28, 34, 44, 62, 68, 76, 82, 92, 104, 110, 118, 128, 134, 166, 184, 202, 212, 266, 286, 296, 314, 328, 350, 356, 376, 406, 428, 436, 460, 470, 506, 520, 532, 562, 580, 638, 650, 652, 680, 692, 722, 734, 740, 778, 812, 820, 824, 862, 896, 908

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..5860

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), this sequence (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)...

Select[Range[0,1000],PrimeQ[#^4+3]&] (* Harvey P. Dale, Aug 02 2023 *)

isok(n, k=4) = isprime(n^k + k - 1); \\ Michel Marcus, Oct 11 2013

A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

Original entry on oeis.org

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840

Offset: 1

Labos Elemer, Apr 23 2002

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008
Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012
Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012
{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

			150 is a term since 149, 151 and 22501 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014

Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014

A199307 Primes of the form 4n^3 + 1.

Original entry on oeis.org

5, 109, 257, 1373, 2917, 4001, 27437, 62501, 157217, 202613, 237277, 296353, 470597, 629857, 665501, 1492993, 1556069, 1898209, 2456501, 2634013, 3217429, 3322337, 4244833, 5038849, 5180117, 6572129, 10512289, 11453153, 12706093

Offset: 1

N. J. A. Sloane, Nov 05 2011

Dirichlet's theorem on primes in arithmetic progressions tells us, for example, that there are infinitely many primes of the form 4n+1. For primes represented by polynomials of degree greater than 1, the Bateman-Horn paper gives a conjecture on the density.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, 16 (1962), 363-367.
Wikipedia, Bateman-Horn Conjecture

Cf. A115104, A001912, A002496, A005574.

[ a: n in [0..200] | IsPrime(a) where a is 4*n^3+1 ]; // Vincenzo Librandi, Nov 08 2011

Select[Table[4n^3+1,{n,1100}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

for(n=1,1e3,if(isprime(t=4*n^3+1),print1(t", "))) \\ Charles R Greathouse IV, Nov 21 2011

A331941 Hardy-Littlewood constant for the polynomial x^2 + 1.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Feb 02 2020

			0.686406731409123004556096348363509434089166550627879778968117...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).

Cf. A002496, A005574, A083844, A199401, A206709, A221712.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1)/2 after setting the required precision.

Equals (1/2)*Product_{p=primes} (1 - Kronecker(-4,p)/(p - 1)).

Equals A199401/2.

cp's OEIS Frontend

A114271 Numbers k such that k^2 + 8 is prime.

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A103854 Positive integers n such that n^6 + 1 is semiprime.

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A096012 Numbers k such that k^2+1 and (k+2)^2+1 are both prime; twin k^2+1 primes.

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A113536 Numbers k such that k^2 + 13 is prime.

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A125260 Numbers k such that k^5 + 4 is prime.

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A035092 Smallest k such that (n^2)*k + 1 is prime.

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A125259 Numbers k such that k^4 + 3 is prime.

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A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

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