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Search: id:A133740
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| A133740 |
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Primes which are the sum of four positive 4th powers. |
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+0
2
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| 19, 179, 419, 499, 643, 673, 769, 883, 1153, 1409, 1459, 1889, 2003, 2083, 2131, 2579, 2609, 2659, 2689, 2819, 3169, 3779, 3889, 3907, 4099, 4129, 4259, 4339, 4513, 4723, 4993, 5009, 5059, 5233, 5347, 5443, 5683, 6529, 6659, 6689, 6899, 7219, 7283, 7459
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Every positive integer is expressible as a sum of (at most) g(4) = 19 biquadratic numbers (Waring's problem). Davenport (1939) showed that G(4) = 16, meaning that all sufficiently large integers require only 16 biquadratic numbers.
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LINKS
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Eric Weisstein's World of Mathematics, Biquadratic Number.
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FORMULA
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{primes} INTERSECTION {a^4 + b^4 + c^4 + d^4} = A000040 INTERSECTION {A000583(a) + A000583(b) + A000583(c) + A000583(d) + for a,b,c,d > 0}
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EXAMPLE
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a(1) = 19 = 2^4 + 1^4 + 1^4 + 1^4 = 16 + 1 + 1 + 1.
a(2) = 179 = 3^4 + 3^4 + 2^4 + 1^4 = 81 + 81 + 16 + 1.
a(3) = 4^4 + 3^4 + 3^4 + 1^4 = 256 + 81 + 81 + 1.
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MATHEMATICA
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Select[Union[ Flatten[Table[ a^4 + b^4 + c^4 + d^4, {a, 1, 10}, {b, 1, a}, {c, 1, b}, {d, 1, c}]]], PrimeQ]
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CROSSREFS
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Cf. A000040, A000583, A003337, A085318.
Sequence in context: A022614 A060222 A041690 this_sequence A125382 A126540 A008419
Adjacent sequences: A133737 A133738 A133739 this_sequence A133741 A133742 A133743
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 31 2007
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