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A002646
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Half-quartan primes: primes of the form p = (x^4 + y^4)/2.
(Formerly M5276 N2294)
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2
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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; internal format)
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OFFSET
| 1,1
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COMMENTS
| The 1001-digit number ((10^250+5659)^4+(10^250+5661)^4)/2, is currently the largest known half-quartan prime. - Paul Muljadi, March 3, 2011
The largest known is now ((2*3960926^2048+1)^4+1^4)/2 with 54051 digits. - Jens Kruse Andersen, March 20, 2011
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REFERENCES
| A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics, 36, 11 (1907), pp. 145-174.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
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MAPLE
| for m from 1 to 15 do for n from 1 to m do a := ((2*m-1)^4+(2*n-1)^4)/2: if isprime(a) then print(m, n, a) fi od od;
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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-François Alcover, Mar 23 2011 *)
Sort[Select[Total/@(Union[Sort/@Tuples[Range[0, 50], 2]]^4)/2, PrimeQ]] (* From Harvey P. Dale, Feb 12 2012 *)
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CROSSREFS
| Sequence in context: A090833 A154577 A201043 * A175110 A096170 A121671
Adjacent sequences: A002643 A002644 A002645 * A002647 A002648 A002649
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KEYWORD
| nonn,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Len Smiley (smiley(AT)math.uaa.alaska.edu)
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