

A133740


Primes which are the sum of four positive 4th powers.


1



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
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OFFSET

1,1


COMMENTS

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.


LINKS

Table of n, a(n) for n=1..44.
Eric Weisstein's World of Mathematics, Biquadratic Number.


FORMULA

{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}


EXAMPLE

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.


MATHEMATICA

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]


CROSSREFS

Cf. A000040, A000583, A003337, A085318.
Sequence in context: A041690 A217698 A172642 * A125382 A126540 A008419
Adjacent sequences: A133737 A133738 A133739 * A133741 A133742 A133743


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Dec 31 2007


STATUS

approved



