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 A000414 #37 Mar 18 2025 20:16:16
%S A000414 4,7,10,12,13,15,16,18,19,20,21,22,23,25,26,27,28,30,31,33,34,35,36,
%T A000414 37,38,39,40,42,43,44,45,46,47,48,49,50,51,52,53,54,55,57,58,59,60,61,
%U A000414 62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81
%N A000414 Numbers that are the sum of 4 nonzero squares.
%C A000414 As the order of addition doesn't matter we can assume terms are in increasing order. - _David A. Corneth_, Aug 01 2020
%H A000414 David A. Corneth, <a href="/A000414/b000414.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A000414 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A000414 a(n) = n + O(log n). - _Charles R Greathouse IV_, Sep 03 2014
%e A000414 From _David A. Corneth_, Aug 01 2020: (Start)
%e A000414 1608 is in the sequence as 1608 = 18^2 + 20^2 + 20^2 + 22^2.
%e A000414 2140 is in the sequence as 2140 = 21^2 + 21^2 + 23^2 + 27^2.
%e A000414 3298 is in the sequence as 3298 = 25^2 + 26^2 + 29^2 + 34^2. (End)
%t A000414 q=16;lst={};Do[Do[Do[Do[z=a^2+b^2+c^2+d^2;If[z<=(q^2)+3,AppendTo[lst,z]],{d,q}],{c,q}],{b,q}],{a,q}];Union@lst (*_Vladimir Joseph Stephan Orlovsky_, Feb 07 2010 *)
%t A000414 Total/@Tuples[Range[10]^2,4]//Union (* _Harvey P. Dale_, Mar 18 2025 *)
%o A000414 (PARI) is(n)=my(k=if(n,n/4^valuation(n,4),2)); k!=2 && k!=6 && k!=14 && !setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ _Charles R Greathouse IV_, Sep 03 2014
%o A000414 (Python)
%o A000414 limit = 10026 # 10000th term in b-file
%o A000414 from functools import lru_cache
%o A000414 nzs = [k*k for k in range(1, int(limit**.5)+2) if k*k + 3 <= limit]
%o A000414 nzss = set(nzs)
%o A000414 @lru_cache(maxsize=None)
%o A000414 def ok(n, m): return n in nzss if m == 1 else any(ok(n-s, m-1) for s in nzs)
%o A000414 print([n for n in range(4, limit+1) if ok(n, 4)]) # _Michael S. Branicky_, Apr 07 2021
%o A000414 (Python)
%o A000414 from itertools import count, islice
%o A000414 def A000414_gen(startvalue=0): # generator of terms >= startvalue
%o A000414 return filter(lambda n:not(n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2,6,14}),count(max(startvalue,0)))
%o A000414 A000414_list = list(islice(A000414_gen(),30)) # _Chai Wah Wu_, Jul 09 2022
%Y A000414 Cf. A000534 (complement).
%Y A000414 A###### (x, y): Numbers that are the form of x nonzero y-th powers.
%Y A000414 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 A000414 nonn,easy
%O A000414 1,1
%A A000414 _N. J. A. Sloane_ and _J. H. Conway_
%E A000414 corrected 6/95