The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A345968 Numbers whose square can be represented in exactly three ways as the sum of a positive square and a positive fourth power. 9
 1625, 6500, 14625, 18785, 24505, 26000, 40625, 58500, 75140, 79625, 88985, 98020, 104000, 120250, 131625, 162500, 169065, 196625, 220545, 234000, 274625, 296225, 300560, 318500, 355940, 365625, 392080, 416000, 481000, 526500, 547230, 586625, 611585, 612625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. No term can be a square (see the comment from Altug Alkan in A111925). Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144). 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. The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. LINKS Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 Karl-Heinz Hofmann, All valid {z,x1,y1,x2,y2,x3,y3} sets up to 10^8 EXAMPLE 29640^2 + 39^4 = 29679^2; 29679 is not a term (only 1 solution). 60^2 + 5^4 = 63^2 + 4^4 = 65^2; 65 is not a term (only 2 solutions). 572^2 + 39^4 = 1500^2 + 25^4 = 1575^2 + 20^4 = 1625^2; 1625 is a term (3 solutions). 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). CROSSREFS 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). Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)). Sequence in context: A022060 A107524 A097225 * A186846 A198510 A241493 Adjacent sequences: A345965 A345966 A345967 * A345969 A345970 A345971 KEYWORD nonn AUTHOR Karl-Heinz Hofmann, Jun 30 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 21 02:42 EDT 2024. Contains 373535 sequences. (Running on oeis4.)