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Search: id:A133750
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| A133750 |
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Primes which are the sum of five positive 4th powers. |
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+0
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| 5, 659, 709, 739, 929, 1283, 1409, 1493, 1523, 1877, 1907, 2099, 2179, 2339, 2689, 2803, 3109, 3187, 3299, 3539, 3733, 3923, 4339, 4357, 5009, 5059, 5443, 5683, 5939, 5987, 6053, 6133, 6529, 7219, 7459, 7829, 8419, 8609, 8819, 8849, 9043, 9539, 10067
(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 + e^4} = A000040 INTERSECTION {A000583(a) + A000583(b) + A000583(c) + A000583(d) + A000583(e) for a,b,c,d,e > 0}
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EXAMPLE
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a(1) = 5 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 = 1 + 1 + 1 + 1 + 1.
a(2) = 659 = 5^4 + 2^4 + 2^4 + 1^4 + 1^4 = 625 + 16 + 16 + 1 + 1.
a(3) = 709 = 5^4 + 3^4 + 1^4 + 1^4 + 1^4 = 625 + 81 + 1 + 1 + 1.
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MATHEMATICA
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Select[Union[ Flatten[Table[ a^4 + b^4 + c^4 + d^4 + e^4, {a, 1, 8}, {b, 1, a}, {c, 1, b}, {d, 1, c}, {e, 1, d}]]], PrimeQ]
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CROSSREFS
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Cf. A000040, A000583, A003337, A085318.
Sequence in context: A068421 A142535 A117709 this_sequence A090947 A000367 A092133
Adjacent sequences: A133747 A133748 A133749 this_sequence A133751 A133752 A133753
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Dec 31 2007
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