A133750 Primes which are the sum of five positive 4th powers.

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, 7349, 7459, 7699, 7829, 8419, 8609, 8819, 8849, 9043, 9539

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 + e^4} = A000040 INTERSECTION {A000583(a) + A000583(b) + A000583(c) + A000583(d) + A000583(e) for a,b,c,d,e > 0}

Example

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.

Mathematica

t = Range[9]^4; Select[Union[Plus @@@ Tuples[t, 5]], # < 10^4 && PrimeQ[#] &] (* Giovanni Resta, Jun 20 2016 *)

Crossrefs

Cf. A000040, A000583, A003337, A085318.

Sequence in context: A117709 A185820 A332165 * A203925 A198597 A180315

Adjacent sequences: A133747 A133748 A133749 * A133751 A133752 A133753

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 31 2007

Extensions

Data corrected by Giovanni Resta, Jun 20 2016

Status

approved