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A345645 Numbers whose square can be represented in exactly one way as the sum of a square and a biquadrate (fourth power).

%I A345645 #74 Nov 22 2021 02:26:22
%S A345645 5,15,20,34,39,41,45,60,80,85,111,125,135,136,150,156,164,175,180,194,
%T A345645 219,240,245,255,265,306,313,320,325,340,351,353,369,371,375,405,410,
%U A345645 444,445,455,500,505,514,540,544,600,605,609,624,629,656,671,674,689
%N A345645 Numbers whose square can be represented in exactly one way as the sum of a square and a biquadrate (fourth power).
%C A345645 Numbers z such that there is exactly one solution to z^2 = x^2 + y^4.
%C A345645 From _Karl-Heinz Hofmann_, Oct 21 2021: (Start)
%C A345645 No term can be a square (see the comment from Altug Alkan in A111925).
%C A345645 Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
%C A345645 Additionally, if the terms have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
%C A345645 The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. (End)
%H A345645 Karl-Heinz Hofmann, <a href="/A345645/b345645.txt">Table of n, a(n) for n = 1..10000</a>
%e A345645 3^2 + 2^4 = 9 + 16 = 25 = 5^2, so 5 is a term.
%e A345645 60^2 + 5^4 = 63^2 + 4^4 = 65^2, so 65 is not a term.
%t A345645 Select[Range@100,Length@Solve[x^2+y^4==#^2&&x>0&&y>0,{x,y},Integers]==1&] (* _Giorgos Kalogeropoulos_, Jun 25 2021 *)
%o A345645 (Python)
%o A345645 terms = []
%o A345645 for i in range(1, 700):
%o A345645     occur = 0
%o A345645     ii = i*i
%o A345645     for j in range(1, i):
%o A345645         k = int((ii - j*j) ** 0.25)
%o A345645         if k*k*k*k + j*j == ii:
%o A345645             occur += 1
%o A345645     if occur == 1:
%o A345645         terms.append(i)
%o A345645 print(terms)
%o A345645 (PARI) inlist(list, v) = for (i=1, #list, if (list[i]==v, return(1)));
%o A345645 isok(m) = {my(list = List()); for (k=1, sqrtnint(m^2, 4), if (issquare(j=m^2-k^4) && !inlist(vecsort([k^4,j^2])), listput(list, vecsort([k^4,j^2])));); #list == 1;} \\ _Michel Marcus_, Jun 26 2021
%Y A345645 Cf. A000290, A000583, A180241, A271576 (all solutions).
%Y A345645 Cf. A345700 (2 solutions), A345968 (3 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions), A346115 (the least solutions).
%Y A345645 Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
%K A345645 nonn
%O A345645 1,1
%A A345645 _Mohammad Tejabwala_, Jun 21 2021