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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)