A003336 Numbers that are the sum of 2 positive 4th powers.
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
Offset: 1
Examples
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)
Links
- Sean A. Irvine, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..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]
- Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
- Eric Weisstein's World of Mathematics, Biquadratic Number.
Crossrefs
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).
Programs
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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 *) -
PARI
list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1,4), for(y=1, min(sqrtnint(lim-x^4,4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024
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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
Formula
{i: A216284(i) > 0}. - R. J. Mathar, Jun 04 2021
Comments