|
%I M5042 N2178 #84 Oct 28 2023 12:22:15
%S 2,17,97,257,337,641,881,1297,2417,2657,3697,4177,4721,6577,10657,
%T 12401,14657,14897,15937,16561,28817,38561,39041,49297,54721,65537,
%U 65617,66161,66977,80177,83537,83777,89041,105601,107377,119617,121937
%N Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.
%C The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144 + 1 = (145310^65536)^4 + 1^4, found by Ricky L Hubbard. - _Jens Kruse Andersen_, Mar 20 2011
%C Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - _Thomas Ordowski_, Feb 22 2017
%D A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
%D N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 22-8 Mu Alpha Theta 1984.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Zak Seidov, <a href="/A002645/b002645.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A. J. C. Cunningham, <a href="/wiki/File:High_quartan_factorisations_and_primes.pdf">High quartan factorisations and primes</a>, Messenger of Mathematics 36 (1907), pp. 145-174.
%H A. J. C. Cunningham, <a href="/A001912/a001912.pdf">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%F A000040 INTERSECTION A003336. - _Jonathan Vos Post_, Sep 23 2006
%F A256852(A049084(a(n))) > 1 for n > 1. - _Reinhard Zumkeller_, Apr 11 2015
%e a(1) = 2 = 1^4 + 1^4.
%e a(2) = 17 = 1^4 + 2^4.
%e a(3) = 97 = 2^4 + 3^4.
%e a(4) = 257 = 1^4 + 4^4.
%t nn = 100000; Sort[Reap[Do[n = a^4 + b^4; If[n <= nn && PrimeQ[n], Sow[n]], {a, nn^(1/4)}, {b, a}]][[2, 1]]]
%t With[{nn=20},Select[Union[Flatten[Table[x^4+y^4,{x,nn},{y,nn}]]],PrimeQ[ #] && #<=nn^4+1&]] (* _Harvey P. Dale_, Aug 10 2021 *)
%o (PARI) upto(lim)=my(v=List(2),t);forstep(x=1,lim^.25,2,forstep(y=2,(lim-x^4)^.25,2,if(isprime(t=x^4+y^4),listput(v,t))));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 05 2011
%o (PARI) list(lim)=my(v=List([2]),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; forstep(y=1+x%2,min(sqrtnint(lim-x4,4), x-1),2, if(isprime(t=x4+y^4), listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Aug 20 2017
%o (Haskell)
%o a002645 n = a002645_list !! (n-1)
%o a002645_list = 2 : (map a000040 $ filter ((> 1) . a256852) [1..])
%o -- _Reinhard Zumkeller_, Apr 11 2015
%Y Subsequence of A002313 and of A028916.
%Y Intersection of A004831 and A000040.
%Y Cf. A002646, A000583, A003336, A000290, A256852.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002
|
