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%I #41 Sep 07 2023 14:13:40
%S 2,4,10,14,74,94,130,134,146,160,230,256,326,340,350,406,430,440,470,
%T 584,634,686,700,704,784,860,920,986,1054,1070,1156,1210,1324,1340,
%U 1354,1366,1394,1420,1456,1460,1564,1700,1784,1816,1876,2006,2080,2096,2174
%N Numbers k such that k^2 + 1 and k^2 + 3 are both prime.
%C 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).
%C From _Jacques Tramu_, Sep 10 2018: (Start)
%C a(10000) = 2473624; C = 2.91596513
%C a(100000) = 35866246; C = 2.70591741
%C a(1000000) = 483764726; C = 2.53454683
%C a(2000000) = 1049178316; C = 2.49209641
%C a(3000000) = 1647417724; C = 2.46880647
%C a(4000000) = 2267125384; C = 2.45259161
%C a(5000000) = 2903162576; C = 2.44036006
%C a(6000000) = 3551848640; C = 2.43024082
%C a(7000000) = 4212006124; C = 2.42214552
%C a(8000000) = 4881390700; C = 2.41510010
%C a(9000000) = 5559542740; C = 2.40915933
%C a(10000000) = 6245573750; C = 2.40405768
%C a(20000000) = 13393786900; C = 2.36959294
%C a(30000000) = 20908970800; C = 2.35131696
%C a(40000000) = 28659267134; C = 2.33835867
%C a(50000000) = 36590858294; C = 2.32865934
%C C is the quotient a(n) / (n * log(n) * log(n)). (End)
%D P. Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, 1996, p. 408.
%H Zak Seidov, <a href="/A080149/b080149.txt">Table of n, a(n) for n = 1..32898</a> (terms < 10^7, first 1000 terms from T. D. Noe)
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
%F Conjecture: a(n) is asymptotic to c*n*log(n)^2 with c around 2.9... - _Benoit Cloitre_, Apr 16 2004
%e 10 is in this sequence because 101 and 103 are both prime.
%t lst={}; Do[If[PrimeQ[m^2+1]&&PrimeQ[m^2+3], AppendTo[lst, m]], {m, 3000}]; lst
%t okQ[n_]:=Module[{n2=n^2},PrimeQ[n2+1]&&PrimeQ[n2+3]]; Select[Range[2200], okQ] (* _Harvey P. Dale_, Apr 21 2011 *)
%t Select[Range[2500],AllTrue[#^2+{1,3},PrimeQ]&] (* _Harvey P. Dale_, Sep 07 2023 *)
%o (PARI) isA080149(n) = isprime(n^2+1) && isprime(n^2+3) \\ _Michael B. Porter_, Mar 22 2010
%Y Cf. A005574, A049422, A145824.
%K easy,nonn
%O 1,1
%A _T. D. Noe_, Jan 30 2003
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