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Numbers that are the sum of 4 nonzero 4th powers.
40

%I #42 May 17 2023 10:41:48

%S 4,19,34,49,64,84,99,114,129,164,179,194,244,259,274,289,304,324,339,

%T 354,369,419,434,499,514,529,544,594,609,628,643,658,673,674,708,723,

%U 738,769,784,788,803,849,868,883,898,913,963,978,1024,1043,1138,1153,1218

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

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

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

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

%e 53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4.

%e 81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4.

%e 106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (End)

%p # returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d.

%p A003338 := proc(n)

%p local a,i,j,k,l,res ;

%p a := 0 ;

%p for i from 1 do

%p if i^4 > n then

%p break ;

%p end if;

%p for j from i do

%p if i^4+j^4 > n then

%p break ;

%p end if;

%p for k from j do

%p if i^4+j^4+k^4> n then

%p break;

%p end if;

%p res := n-i^4-j^4-k^4 ;

%p if issqr(res) then

%p res := sqrt(res) ;

%p if issqr(res) then

%p l := sqrt(res) ;

%p if l >= k then

%p a := a+1 ;

%p end if;

%p end if;

%p end if;

%p end do:

%p end do:

%p end do:

%p a ;

%p end proc:

%p for n from 1 do

%p if A003338(n) > 0 then

%p print(n) ;

%p end if;

%p end do: # _R. J. Mathar_, May 17 2023

%t f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]],seq}, seq=Union[Total/@(Tuples[Range[nn],{4}]^4)]; Select[seq,#<=maxno&]]

%t f[1000] (* _Harvey P. Dale_, Feb 27 2011 *)

%o (Python)

%o limit = 1218

%o from functools import lru_cache

%o qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]

%o qds = set(qd)

%o @lru_cache(maxsize=None)

%o def findsums(n, m):

%o if m == 1: return {(n, )} if n in qds else set()

%o return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))

%o print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # _Michael S. Branicky_, Apr 19 2021

%Y Cf. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero 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).

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_