This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003338 #43 Feb 16 2025 08:32:27
%S A003338 4,19,34,49,64,84,99,114,129,164,179,194,244,259,274,289,304,324,339,
%T A003338 354,369,419,434,499,514,529,544,594,609,628,643,658,673,674,708,723,
%U A003338 738,769,784,788,803,849,868,883,898,913,963,978,1024,1043,1138,1153,1218
%N A003338 Numbers that are the sum of 4 nonzero 4th powers.
%C A003338 As the order of addition doesn't matter we can assume terms are in nondecreasing order. - _David A. Corneth_, Aug 01 2020
%H A003338 David A. Corneth, <a href="/A003338/b003338.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A003338 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number.</a>
%e A003338 From _David A. Corneth_, Aug 01 2020: (Start)
%e A003338 53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4.
%e A003338 81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4.
%e A003338 106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (End)
%p A003338 # returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d.
%p A003338 A003338 := proc(n)
%p A003338 local a,i,j,k,l,res ;
%p A003338 a := 0 ;
%p A003338 for i from 1 do
%p A003338 if i^4 > n then
%p A003338 break ;
%p A003338 end if;
%p A003338 for j from i do
%p A003338 if i^4+j^4 > n then
%p A003338 break ;
%p A003338 end if;
%p A003338 for k from j do
%p A003338 if i^4+j^4+k^4> n then
%p A003338 break;
%p A003338 end if;
%p A003338 res := n-i^4-j^4-k^4 ;
%p A003338 if issqr(res) then
%p A003338 res := sqrt(res) ;
%p A003338 if issqr(res) then
%p A003338 l := sqrt(res) ;
%p A003338 if l >= k then
%p A003338 a := a+1 ;
%p A003338 end if;
%p A003338 end if;
%p A003338 end if;
%p A003338 end do:
%p A003338 end do:
%p A003338 end do:
%p A003338 a ;
%p A003338 end proc:
%p A003338 for n from 1 do
%p A003338 if A003338(n) > 0 then
%p A003338 print(n) ;
%p A003338 end if;
%p A003338 end do: # _R. J. Mathar_, May 17 2023
%t A003338 f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]],seq}, seq=Union[Total/@(Tuples[Range[nn],{4}]^4)]; Select[seq,#<=maxno&]]
%t A003338 f[1000] (* _Harvey P. Dale_, Feb 27 2011 *)
%o A003338 (Python)
%o A003338 limit = 1218
%o A003338 from functools import lru_cache
%o A003338 qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
%o A003338 qds = set(qd)
%o A003338 @lru_cache(maxsize=None)
%o A003338 def findsums(n, m):
%o A003338 if m == 1: return {(n, )} if n in qds else set()
%o A003338 return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))
%o A003338 print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # _Michael S. Branicky_, Apr 19 2021
%Y A003338 Cf. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero powers).
%Y A003338 A###### (x, y): Numbers that are the form of x nonzero y-th powers.
%Y A003338 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).
%K A003338 nonn,easy
%O A003338 1,1
%A A003338 _N. J. A. Sloane_