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%I M5276 N2294 #77 Sep 02 2023 18:05:27
%S 41,313,353,1201,3593,4481,7321,8521,10601,14281,14321,14593,21601,
%T 26513,32633,41761,41801,42073,42961,49081,56041,66361,67073,72481,
%U 90473,97241,97553,104561,106921,111521,139921,141121,165233,195353,198593
%N Half-quartan primes: primes of the form p = (x^4 + y^4)/2.
%C The 1001-digit number ((10^250 + 5659)^4 + (10^250 + 5661)^4)/2 is currently the largest known half-quartan prime. - _Paul Muljadi_, Mar 03 2011
%C The largest known is now ((2*3960926^2048 + 1)^4 + 1^4)/2 with 54051 digits. - _Jens Kruse Andersen_, Mar 20 2011
%C Primes of the form p = a^2 + b^2 with a > b > 0 such that a + b and a - b are squares. - _Thomas Ordowski_, Jul 07 2016
%C Primes p = a^2 + b^2 with a > b > 0 such that a^2 - b^2 is a square. - _Thomas Ordowski_, Feb 14 2017
%C Primes p > 5 such that the Diophantine equation X^4 + Y^2 = p^2 has a solution X,Y with nonzero X. Then X must be odd. - _Thomas Ordowski_ and _Robert G. Wilson v_, Nov 29 2017
%D A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
%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 T. D. Noe, <a href="/A002646/b002646.txt">Table of n, a(n) for n = 1..10000</a>
%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 Tim Evink, Jaap Top, and Jakob Dirk Top, <a href="https://arxiv.org/abs/2105.01450">A remark on prime (non)congruent numbers</a>, arXiv:2105.01450 [math.NT], 2021. See p. 12.
%e 41 is in the sequence since it is prime and 41 = (3^4 + 1^4)/2. - _Michael B. Porter_, Jul 07 2016
%p N:= 10^6: # to get all terms <= N
%p sort(select(isprime, convert({seq(seq((x^4+y^4)/2, y=x..floor((2*N-x^4)^(1/4)),2),x=1..floor((2*N-1)^(1/4)),2)},list))); # _Robert Israel_, Jul 11 2016
%t nmax = 200000; jmax = Floor[(nmax/8)^(1/4)]; s = {}; Do[n = ((2 j + 1)^4 + (2 k + 1)^4)/2; If[n <= nmax && PrimeQ[n], AppendTo[s, n]], {j, 0, jmax}, {k, j, jmax}]; Union[s] (* _Jean-François Alcover_, Mar 23 2011 *)
%t Sort[Select[Total/@(Union[Sort/@Tuples[Range[0,50],2]]^4)/2,PrimeQ]] (* _Harvey P. Dale_, Feb 12 2012 *)
%o (Haskell)
%o a002646 n = a002646_list !! (n-1)
%o a002646_list = [hqp | x <- [1, 3 ..], y <- [1, 3 .. x - 1],
%o let hqp = div (x ^ 4 + y ^ 4) 2, a010051' hqp == 1]
%o -- _Reinhard Zumkeller_, Jul 15 2013
%Y Cf. A000583, A002645, A005408, A010051.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Len Smiley_
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