cp's OEIS Frontend

A271576 Integers whose square is of the form x^2 + y^4, with x,y > 0 (see A111925).

%I A271576 #43 Nov 16 2021 12:00:22
%S A271576 5,15,20,34,39,41,45,60,65,80,85,111,125,135,136,145,150,156,164,175,
%T A271576 180,194,219,240,245,255,260,265,306,313,320,325,340,351,353,369,371,
%U A271576 375,405,410,444,445,455,500,505,514,540,544,580,585,600,605,609,624,629,656,671,674,689
%N A271576 Integers whose square is of the form x^2 + y^4, with x,y > 0 (see A111925).
%C A271576 If z is in this sequence, z*k^2 is, for k > 0. Also note that since there are no fourth powers in A111925, there are no squares in this sequence. - _Altug Alkan_, Apr 10 2016
%C A271576 From _Karl-Heinz Hofmann_, Oct 22 2021: (Start)
%C A271576 Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
%C A271576 If the terms additionally 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. (End)
%C A271576 From _Jon E. Schoenfield_, Nov 15 2021: (Start)
%C A271576 Apparently, the vast majority of the terms of this sequence can be expressed as x^2 + y^4 with x,y > 0 in only one way (A345645 lists those terms), but some can be so expressed in exactly two, three, four, five, six, or more ways. Among the first 3976926961 terms of this sequence,
%C A271576         3948648229 are in A345645 (exactly 1 way),
%C A271576           25415062 are in A345700 (exactly 2 ways),
%C A271576            2697713 are in A345968 (exactly 3 ways),
%C A271576             161543 are in A346110 (exactly 4 ways),
%C A271576               3989 are in A348655 (exactly 5 ways),
%C A271576                424 are in A349324 (exactly 6 ways),
%C A271576         and just 1 -- a(3976926961) = 2474052064291275 = A346115(7) -- is a number whose square can be written as x^2 + y^4 with x,y > 0 in exactly 7 ways. The ratios of successive counts above, i.e., 3948648229/25415062, 25415062/2697713, 2697713/161543, 161543/3989, 3989/424, and 424/1, are approximately 155.4, 9.4, 16.7, 40.5, 9.4, and 424.0. What is it that (over the interval [1, 2474052064291275], at least) makes numbers whose squares can be written as x^2 + y^4 with x,y > 0 in more than 6 ways so much rarer than those that can be written thus in exactly 6 ways? (End)
%H A271576 Chai Wah Wu, <a href="/A271576/b271576.txt">Table of n, a(n) for n = 1..10000</a>
%e A271576 5^2 = 25 = 9 + 16 = 3^2 + 2^4, so 5 is a term.
%t A271576 Select[Range@ 200, Resolve[Exists[{x, y}, Reduce[#^2 == x^2 + y^4, {x, y}, Integers], And[x > 0, y > 0]]] &] (* _Michael De Vlieger_, Apr 10 2016 *)
%o A271576 (PARI) isok(n) = n = n^2; for (k=1, n-1, if (issquare(k) && (p=ispower(n-k, 4)), return (1)))
%o A271576 (PARI) is(n)=my(n2=n^2); for(b=sqrtnint(2*n-2,4)+1, sqrtint(n-1), if(issquare(n2-b^4), return(1))); 0 \\ _Charles R Greathouse IV_, Nov 16 2021
%Y A271576 Cf. A111925, A271577, A345645 (in exactly 1 way).
%Y A271576 Cf. A345700 (in exactly 2 ways), A345968 (in exactly 3 ways).
%Y A271576 Cf. A346110 (in exactly 4 ways), A349324 (in exactly 6 ways), A346115 (the least solutions).
%Y A271576 Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
%K A271576 nonn
%O A271576 1,1
%A A271576 _Michel Marcus_, Apr 10 2016