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A365657 Integers k such that k^2 can be written as the sum of three positive fourth powers.

%I A365657 #43 Sep 28 2023 22:21:24
%S A365657 481,1924,4329,7696,12025,17316,23569,24961,28721,30784,38961,48100,
%T A365657 58201,65441,69121,69264,81289,94276,99844,108225,113241,114884,
%U A365657 123136,139009,155844,173641,192400,212121,224649,232804,254449,258489,261764,276484,277056,300625,325156
%N A365657 Integers k such that k^2 can be written as the sum of three positive fourth powers.
%C A365657 Primitive solutions are in A365688.
%C A365657 From _Jon E. Schoenfield_, Sep 15 2023: (Start)
%C A365657 If k is a term, then so is m^2 * k for every m > 1.
%C A365657 Every even term is four times a smaller term.
%C A365657 Every odd term is the square root of the sum of one odd fourth power and two even fourth powers.
%C A365657 (End)
%D A365657 Jean-Marie De Koninck, "Those Fascinating Numbers", AMS, 2008, entry 481.
%e A365657 1924^2 = 24^4 + 30^4 + 40^4.
%o A365657 (Python)
%o A365657 from itertools import count, islice
%o A365657 from sympy import integer_nthroot
%o A365657 def A365657_gen(startvalue=1): # generator of terms >= startvalue
%o A365657     for k in count(max(startvalue,1)):
%o A365657         m, flag = k**2, False
%o A365657         for x in count(1):
%o A365657             if (x4:=x**4)+2>m or flag:
%o A365657                 break
%o A365657             for y in range(min(x,integer_nthroot(m-x4-1,4)[0]),0,-1):
%o A365657                 if (z4:=m-x4-(y4:=y**4))>y4 or flag:
%o A365657                     break
%o A365657                 if integer_nthroot(z4,4)[1]:
%o A365657                     yield k
%o A365657                     flag = True
%o A365657                     break
%o A365657 A365657_list = list(islice(A365657_gen(),6)) # _Chai Wah Wu_, Sep 19 2023
%Y A365657 Cf. A003828, A365688.
%K A365657 nonn
%O A365657 1,1
%A A365657 _Jud McCranie_, Sep 14 2023