cp's OEIS Frontend

A346110 Numbers whose square can be represented in exactly four ways as the sum of a positive square and a positive fourth power.

%I A346110 #57 Jan 01 2025 22:15:13
%S A346110 469625,1878500,2224625,4226625,7514000,8898500,11740625,15289625,
%T A346110 16906500,20021625,23011625,25716665,30056000,35594000,38039625,
%U A346110 46962500,54316275,55615625,56824625,61158500,67626000,79366625,80086500,92046500,92481870
%N A346110 Numbers whose square can be represented in exactly four ways as the sum of a positive square and a positive fourth power.
%C A346110 Terms are numbers z such that there are exactly four solutions to z^2 = x^2 + y^4, where x, y and z belong to the set of positive integers.
%C A346110 Terms cannot be a square (see the comment from Altug Alkan in A111925).
%C A346110 Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
%C A346110 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 A346110 The special prime factor 2 has the same behavior. Means: If the term is even, x and y must be even too.
%C A346110 Apparently, all terms are divisible by 65. The divided terms are in A346594. Are there exceptions for n > 25? - _Hugo Pfoertner_, Jul 14 2021, Jul 29 2021
%C A346110 Yes, there are exceptions: a(44,46,53,95,97) are not divisible by 65 (5*13) but they have in common: They are divisible by 145 (5*29). - _Karl-Heinz Hofmann_, Aug 28 2021
%H A346110 Jon E. Schoenfield, <a href="/A346110/b346110.txt">Table of n, a(n) for n = 1..10000</a> (first 97 terms from Karl-Heinz Hofmann)
%H A346110 Karl-Heinz Hofmann, <a href="/A346110/a346110_1.pdf">All valid {z,x1,y1,x2,y2,x3,y3,x4,y4} sets up to 10^9</a>.
%H A346110 Karl-Heinz Hofmann, <a href="/A346110/a346110.gif">A 3D Animation of the solutions up to 10^9</a>.
%H A346110 Karl-Heinz Hofmann, <a href="/A346110/a346110.py.txt">Python code (not only for 4 Solutions)</a>.
%e A346110 29679^2 = 29640^2 + 39^4, so 29679 is not a term (only one solution).
%e A346110 60^2 + 5^4 = 63^2 + 4^4 = 65^2, so 65 is not a term (only two solutions).
%e A346110 572^2 + 39^4 = 1500^2 + 25^4 = 1575^2 + 20^4 = 1625^2, so 1625 is not a term (only three solutions).
%e A346110 165308^2 + 663^4 = 349575^2 + 560^4 = 433500^2 + 425^4 = 455175^2 + 340^4 = 469625^2, so 469625 is a term (four solutions).
%o A346110 (Python) # See Hofmann link.
%Y A346110 Cf. A000290, A000583.
%Y A346110 Cf. A271576 (all solutions), A345645 (one solution), A345700 (two solutions), A345968 (three solutions), A348655 (five solutions), A349324 (six solutions), A346115 (the least solutions).
%Y A346110 Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
%K A346110 nonn
%O A346110 1,1
%A A346110 _Karl-Heinz Hofmann_, Jul 05 2021