%I #66 Jul 13 2024 21:32:41

%S 2,17,32,82,97,162,257,272,337,512,626,641,706,881,1250,1297,1312,

%T 1377,1552,1921,2402,2417,2482,2592,2657,3026,3697,4097,4112,4177,

%U 4352,4721,4802,5392,6497,6562,6577,6642,6817,7186,7857,8192,8962,10001,10016,10081,10256,10625

%N Numbers that are the sum of 2 positive 4th powers.

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

%C There are no squares in this sequence. - _Altug Alkan_, Apr 08 2016

%C As the order of addition doesn't matter we can assume terms are in nondecreasing order. - _David A. Corneth_, Aug 01 2020

%H Sean A. Irvine, <a href="/A003336/b003336.txt">Table of n, a(n) for n = 1..20000</a> (terms 1..1000 from T. D. Noe, terms 1001..10000 from David A. Corneth)

%H A. Bremner and P. Morton, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002225409">A new characterization of the integer 5906</a>, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a generalized Fermat-Wiles equation</a> [broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]

%H Samuel S. Wagstaff, Jr., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Wagstaff/wagstaff8.html">Equal Sums of Two Distinct Like Powers</a>, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number</a>.

%F {i: A216284(i) > 0}. - _R. J. Mathar_, Jun 04 2021

%e From _David A. Corneth_, Aug 01 2020: (Start)

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

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

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

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

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

%o (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

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

%o is(n)=#thue(T,n)>0 && !issquare(n) \\ _Charles R Greathouse IV_, Feb 26 2017

%o (Python)

%o def aupto(lim):

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

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

%o return sorted(p2)

%o print(aupto(10625)) # _Michael S. Branicky_, Mar 18 2021

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

%Y Cf. A088687 (2 distinct 4th powers).

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

%Y 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).

%Y Cf. A000583 (4th powers).

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_