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A003336 Numbers that are the sum of 2 positive 4th powers. 75
2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8192, 8962, 10001, 10016, 10081, 10256, 10625 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers n such that n = x^4 + y^4 has a solution in positive integers x, y.

There are no squares in this sequence. - Altug Alkan, Apr 08 2016

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe, terms 1001 to 10000 from David A. Corneth)

A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.

S. R. Finch, On a generalized Fermat-Wiles equation [broken link]

Steven R. Finch, On  Generalized Fermat-Wiles Equation [From the Wayback Machine]

Eric Weisstein's World of Mathematics, Biquadratic Number

FORMULA

{i: A216284(i) > 0}. - R. J. Mathar, Jun 04 2021

EXAMPLE

From David A. Corneth, Aug 01 2020: (Start)

16378801 is in the sequence as 16378801 = 43^4 + 60^4.

39126977 is in the sequence as 39126977 = 49^4 + 76^4.

71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)

MATHEMATICA

nn=12; Select[Union[Plus@@@(Tuples[Range[nn], {2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)

Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Apr 08 2016 *)

PROG

(PARI) list(lim)=my(v=List(2)); for(x=1, lim^.25, for(y=1, min((lim-x^4)^.25, x), listput(v, x^4+y^4))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Apr 24 2012

(PARI) T=thueinit('x^4+1, 1);

is(n)=#thue(T, n)>0 && !issquare(n) \\ Charles R Greathouse IV, Feb 26 2017

(Python)

def aupto(lim):

  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)

  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)

  return sorted(p2)

print(aupto(10625)) # Michael S. Branicky, Mar 18 2021

CROSSREFS

5906 is the first term in A060387 but not in this sequence. Cf. A020897.

Cf. A088687 (2 distinct 4th powers).

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Cf. A000583 (4th powers).

Sequence in context: A162622 A078164 A060387 * A344187 A212740 A212742

Adjacent sequences:  A003333 A003334 A003335 * A003337 A003338 A003339

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 24 03:27 EDT 2021. Contains 345415 sequences. (Running on oeis4.)