A003338 Numbers that are the sum of 4 nonzero 4th powers.
4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153, 1218
Offset: 1
Examples
From _David A. Corneth_, Aug 01 2020: (Start) 53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4. 81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4. 106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (End)
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
- Eric Weisstein's World of Mathematics, Biquadratic Number.
Crossrefs
Cf. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
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).
Programs
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Maple
# returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d. A003338 := proc(n) local a,i,j,k,l,res ; a := 0 ; for i from 1 do if i^4 > n then break ; end if; for j from i do if i^4+j^4 > n then break ; end if; for k from j do if i^4+j^4+k^4> n then break; end if; res := n-i^4-j^4-k^4 ; if issqr(res) then res := sqrt(res) ; if issqr(res) then l := sqrt(res) ; if l >= k then a := a+1 ; end if; end if; end if; end do: end do: end do: a ; end proc: for n from 1 do if A003338(n) > 0 then print(n) ; end if; end do: # R. J. Mathar, May 17 2023
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Mathematica
f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]],seq}, seq=Union[Total/@(Tuples[Range[nn],{4}]^4)]; Select[seq,#<=maxno&]] f[1000] (* Harvey P. Dale, Feb 27 2011 *) -
Python
limit = 1218 from functools import lru_cache qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit] qds = set(qd) @lru_cache(maxsize=None) def findsums(n, m): if m == 1: return {(n, )} if n in qds else set() return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1)) print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # Michael S. Branicky, Apr 19 2021
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