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A002646 Half-quartan primes: primes of the form p = (x^4 + y^4)/2.
(Formerly M5276 N2294)
5
41, 313, 353, 1201, 3593, 4481, 7321, 8521, 10601, 14281, 14321, 14593, 21601, 26513, 32633, 41761, 41801, 42073, 42961, 49081, 56041, 66361, 67073, 72481, 90473, 97241, 97553, 104561, 106921, 111521, 139921, 141121, 165233, 195353, 198593 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

The largest known is now ((2*3960926^2048+1)^4+1^4)/2 with 54051 digits. - Jens Kruse Andersen, Mar 20 2011

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

Primes p = a^2 + b^2 with a > b > 0 such that a^2 - b^2 is a square. - Thomas Ordowski, Feb 14 2017

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

REFERENCES

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

EXAMPLE

41 is in the sequence since it is prime and 41 = (3^4 + 1^4)/2. - Michael B. Porter, Jul 07 2016

MAPLE

N:= 10^6: # to get all terms <= N

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

MATHEMATICA

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-Fran├žois Alcover, Mar 23 2011 *)

Sort[Select[Total/@(Union[Sort/@Tuples[Range[0, 50], 2]]^4)/2, PrimeQ]] (* Harvey P. Dale, Feb 12 2012 *)

PROG

(Haskell)

a002646 n = a002646_list !! (n-1)

a002646_list = [hqp | x <- [1, 3 ..], y <- [1, 3 .. x - 1],

                      let hqp = div (x ^ 4 + y ^ 4) 2, a010051' hqp == 1]

-- Reinhard Zumkeller, Jul 15 2013

CROSSREFS

Cf. A000583, A002645, A005408, A010051.

Sequence in context: A074281 A090833 A201043 * A175110 A096170 A277201

Adjacent sequences:  A002643 A002644 A002645 * A002647 A002648 A002649

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Len Smiley

STATUS

approved

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Last modified June 16 11:12 EDT 2019. Contains 324152 sequences. (Running on oeis4.)