A133740 Primes which are the sum of four positive 4th powers.

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

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