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A003336 Numbers that are the sum of 2 positive 4th powers.

%I A003336 #67 Feb 16 2025 08:32:27
%S A003336 2,17,32,82,97,162,257,272,337,512,626,641,706,881,1250,1297,1312,
%T A003336 1377,1552,1921,2402,2417,2482,2592,2657,3026,3697,4097,4112,4177,
%U A003336 4352,4721,4802,5392,6497,6562,6577,6642,6817,7186,7857,8192,8962,10001,10016,10081,10256,10625
%N A003336 Numbers that are the sum of 2 positive 4th powers.
%C A003336 Numbers k such that k = x^4 + y^4 has a solution in positive integers x, y.
%C A003336 There are no squares in this sequence. - _Altug Alkan_, Apr 08 2016
%C A003336 As the order of addition doesn't matter we can assume terms are in nondecreasing order. - _David A. Corneth_, Aug 01 2020
%H A003336 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 A003336 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 A003336 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 A003336 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 A003336 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 A003336 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number</a>.
%F A003336 {i: A216284(i) > 0}. - _R. J. Mathar_, Jun 04 2021
%e A003336 From _David A. Corneth_, Aug 01 2020: (Start)
%e A003336 16378801 is in the sequence as 16378801 = 43^4 + 60^4.
%e A003336 39126977 is in the sequence as 39126977 = 49^4 + 76^4.
%e A003336 71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)
%t A003336 nn=12; Select[Union[Plus@@@(Tuples[Range[nn],{2}]^4)], # <= nn^4&] (* _Harvey P. Dale_, Dec 29 2010 *)
%t A003336 Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, _} -> Nothing] > 0 &] (* _Michael De Vlieger_, Apr 08 2016 *)
%o A003336 (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 A003336 (PARI) T=thueinit('x^4+1,1);
%o A003336 is(n)=#thue(T,n)>0 && !issquare(n) \\ _Charles R Greathouse IV_, Feb 26 2017
%o A003336 (Python)
%o A003336 def aupto(lim):
%o A003336   p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
%o A003336   p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
%o A003336   return sorted(p2)
%o A003336 print(aupto(10625)) # _Michael S. Branicky_, Mar 18 2021
%Y A003336 5906 is the first term in A060387 but not in this sequence. Cf. A020897.
%Y A003336 Cf. A088687 (2 distinct 4th powers).
%Y A003336 A###### (x, y): Numbers that are the form of x nonzero y-th powers.
%Y A003336 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 A003336 Cf. A000583 (4th powers).
%K A003336 nonn,easy
%O A003336 1,1
%A A003336 _N. J. A. Sloane_