cp's OEIS Frontend

A345700 Numbers whose square can be represented in exactly two ways as the sum of a positive square and a positive fourth power.

%I A345700 #60 Dec 10 2021 02:46:28
%S A345700 65,145,260,580,585,1025,1040,1105,1305,2320,2340,2465,3185,3625,4100,
%T A345700 4160,4335,4420,4810,5220,5265,6625,7105,7585,7865,8405,9225,9280,
%U A345700 9360,9860,9945,10985,11745,12740,14500,16400,16465,16640,17340,17545,17680,19240
%N A345700 Numbers whose square can be represented in exactly two ways as the sum of a positive square and a positive fourth power.
%C A345700 Terms are numbers z such that there are exactly two solutions to z^2 = x^2 + y^4, where x, y and z belong to the set of positive integers.
%C A345700 Terms cannot be a square (see the comment from Altug Alkan in A111925).
%C A345700 Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
%C A345700 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.
%C A345700 The lower limit of the ratio x/y is sqrt(2).
%H A345700 Jon E. Schoenfield, <a href="/A345700/b345700.txt">Table of n, a(n) for n = 1..10000</a>
%H A345700 Karl-Heinz Hofmann, <a href="/A345700/a345700.pdf">All valid {z,x1,y1,x2,y2} sets up to 10^7</a>
%H A345700 Karl-Heinz Hofmann, <a href="/A345700/a345700_1.txt">Python code for multiple solutions</a>
%e A345700 29679^2 = 29640^2 + 39^4, so 29679 is not a term (only one solution).
%e A345700 17680^2 = 15088^2 + 96^4 = 17472^2 + 52^4, so 17680 is a term.
%e A345700 36900^2 = 36000^2 + 90^4 = 36828^2 + 48^4, so 36900 is a term.
%e A345700 18785^2 = 18207^2 + 68^4 = 17340^2 + 85^4 = 13983^2 + 112^4, so 18785 is not a term (three solutions).
%o A345700 (Python) # see link above.
%o A345700 (PARI) inlist(list, v) = for (i=1, #list, if (list[i]==v, return(1)));
%o A345700 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 == 2;} \\ _Michel Marcus_, Jun 26 2021
%o A345700 (PARI) is(n)=my(n2=n^2,s); for(y=sqrtnint(2*n-2,4)+1,sqrtint(n-1), if(issquare(n2-y^4) && s++>2, return(0))); s==2; \\ _Charles R Greathouse IV_, Jul 02 2021
%Y A345700 Cf. A111925, A271576, A345645 (in exactly 1 way), A345968 (in exactly 3 ways), A346110 (in exactly 4 ways), A348655 (in exactly 5 ways), A349324 (in exactly 6 ways), A346115 (the least solutions).
%Y A345700 Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
%K A345700 nonn
%O A345700 1,1
%A A345700 _Karl-Heinz Hofmann_, Jun 24 2021