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A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

%I A309763 #11 Feb 16 2025 08:33:55
%S A309763 259,2674,2689,2754,2929,3298,3969,4144,4209,5074,6579,6594,6659,6769,
%T A309763 6834,7203,7874,8194,8979,9154,9234,10113,10674,11298,12673,12913,
%U A309763 13139,14674,14689,14754,16563,16578,16643,16818,17187,17234,17299,17314,17858,18963,19699
%N A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.
%H A309763 Robert Israel, <a href="/A309763/b309763.txt">Table of n, a(n) for n = 1..10000</a>
%H A309763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number</a>
%e A309763 259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4, so 259 is in the sequence.
%p A309763 N:= 10^5: # for terms <= N
%p A309763 V:= Vector(N, datatype=integer[4]):
%p A309763 for a from 1 while a^4 <= N do
%p A309763   for b from 1 to a while a^4+b^4 <= N do
%p A309763     for c from 1 to b while a^4 + b^4+ c^4 <= N do
%p A309763       for d from 1 to c do
%p A309763          v:= a^4+b^4+c^4+d^4;
%p A309763          if v > N then break fi;
%p A309763          V[v]:= V[v]+1
%p A309763 od od od od:
%p A309763 select(i -> V[i]>1, [$1..N]); # _Robert Israel_, Oct 07 2019
%t A309763 Select[Range@20000, Length@Select[PowersRepresentations[#, 4, 4], ! MemberQ[#, 0] &] > 1 &]
%Y A309763 Cf. A000583, A003338, A018786, A025367, A025406, A309762, A344193, A344238, A344241, A344644.
%K A309763 nonn
%O A309763 1,1
%A A309763 _Ilya Gutkovskiy_, Aug 15 2019