cp's OEIS Frontend

A345968 Numbers whose square can be represented in exactly three ways as the sum of a positive square and a positive fourth power.

%I A345968 #34 Nov 22 2021 02:27:07
%S A345968 1625,6500,14625,18785,24505,26000,40625,58500,75140,79625,88985,
%T A345968 98020,104000,120250,131625,162500,169065,196625,220545,234000,274625,
%U A345968 296225,300560,318500,355940,365625,392080,416000,481000,526500,547230,586625,611585,612625
%N A345968 Numbers whose square can be represented in exactly three ways as the sum of a positive square and a positive fourth power.
%C A345968 Terms are numbers z such that there are exactly 3 solutions to z^2 = x^2 + y^4, where x, y and z belong to the set of positive integers.
%C A345968 No term can be a square (see the comment from Altug Alkan in A111925).
%C A345968 Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
%C A345968 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 A345968 The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too.
%H A345968 Jon E. Schoenfield, <a href="/A345968/b345968.txt">Table of n, a(n) for n = 1..10000</a>
%H A345968 Karl-Heinz Hofmann, <a href="/A345968/a345968.pdf">All valid {z,x1,y1,x2,y2,x3,y3} sets up to 10^8</a>
%e A345968 29640^2 + 39^4 = 29679^2; 29679 is not a term (only 1 solution).
%e A345968 60^2 + 5^4 = 63^2 + 4^4 = 65^2; 65 is not a term (only 2 solutions).
%e A345968 572^2 + 39^4 = 1500^2 + 25^4 = 1575^2 + 20^4 = 1625^2; 1625 is a term (3 solutions).
%e A345968 165308^2 + 663^4 = 349575^2 + 560^4 = 433500^2 + 425^4 = 455175^2 + 340^4 = 469625^2; 469625 is not a term (4 solutions).
%Y A345968 Cf. A271576 (1 and more solutions), A345645 (1 solution), A345700 (2 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions), A346115 (the least solutions).
%Y A345968 Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
%K A345968 nonn
%O A345968 1,1
%A A345968 _Karl-Heinz Hofmann_, Jun 30 2021