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

1,1

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)

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.

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A005574, A049422, A145824.

Sequence in context: A078775 A351160 A056392 * A217134 A351927 A333619

Adjacent sequences: A080146 A080147 A080148 * A080150 A080151 A080152

Keyword

easy,nonn

Author

T. D. Noe, Jan 30 2003

Status

approved