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A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime. 6
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 (list; graph; refs; listen; history; text; internal format)
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.

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 Mathematica, Vol. 44, pp. 1-70, 1923.

LINKS

Zak Seidov, Table of n, a(n) for n = 1..32898 (terms < 10^7, first 1000 terms from T. D. Noe)

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 *)

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: A236547 A078775 A056392 * A217134 A333619 A128513

Adjacent sequences:  A080146 A080147 A080148 * A080150 A080151 A080152

KEYWORD

easy,nonn

AUTHOR

T. D. Noe, Jan 30 2003

STATUS

approved

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Last modified June 13 04:23 EDT 2021. Contains 344980 sequences. (Running on oeis4.)