A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

2, 4, 10, 14, 74, 94, 130, 134, 146, 160, 230, 256, 326, 340, 350, 406, 430, 440, 470, 584, 634, 686, 700, 704, 784, 860, 920, 986, 1054, 1070, 1156, 1210, 1324, 1340, 1354, 1366, 1394, 1420, 1456, 1460, 1564, 1700, 1784, 1816, 1876, 2006, 2080, 2096, 2174

Offset: 1

Author

T. D. Noe, Jan 30 2003

Keywords

• easy
• nonn

Comments

Hardy and Littlewood conjecture that this sequence is infinite. This sequence is the intersection of A005574 (k such that k^2 + 1 is prime) and A049422 (k such that k^2 + 3 is prime).

From Jacques Tramu, Sep 10 2018: (Start)

a(10000) = 2473624; C = 2.91596513

a(100000) = 35866246; C = 2.70591741

a(1000000) = 483764726; C = 2.53454683

a(2000000) = 1049178316; C = 2.49209641

a(3000000) = 1647417724; C = 2.46880647

a(4000000) = 2267125384; C = 2.45259161

a(5000000) = 2903162576; C = 2.44036006

a(6000000) = 3551848640; C = 2.43024082

a(7000000) = 4212006124; C = 2.42214552

a(8000000) = 4881390700; C = 2.41510010

a(9000000) = 5559542740; C = 2.40915933

a(10000000) = 6245573750; C = 2.40405768

a(20000000) = 13393786900; C = 2.36959294

a(30000000) = 20908970800; C = 2.35131696

a(40000000) = 28659267134; C = 2.33835867

a(50000000) = 36590858294; C = 2.32865934

C is the quotient a(n) / (n * log(n) * log(n)). (End)

Examples

			10 is in this sequence because 101 and 103 are both prime.
		

References

• P. Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, 1996, p. 408.

Links

• Zak Seidov, Table of n, a(n) for n = 1..32898 (terms < 10^7, first 1000 terms from T. D. Noe)
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Crossrefs

Cf. A005574, A049422, A145824.

Programs

Mathematica

lst={}; Do[If[PrimeQ[m^2+1]&&PrimeQ[m^2+3], AppendTo[lst, m]], {m, 3000}]; lst
okQ[n_]:=Module[{n2=n^2},PrimeQ[n2+1]&&PrimeQ[n2+3]]; Select[Range[2200], okQ]  (* Harvey P. Dale, Apr 21 2011 *)
Select[Range[2500],AllTrue[#^2+{1,3},PrimeQ]&] (* Harvey P. Dale, Sep 07 2023 *)

PARI

isA080149(n) = isprime(n^2+1) && isprime(n^2+3) \\ Michael B. Porter, Mar 22 2010

Formula

Conjecture: a(n) is asymptotic to c*n*log(n)^2 with c around 2.9... - Benoit Cloitre, Apr 16 2004