

A349663


Positive numbers x for which x^2 can be expressed as z^2  y^4 with y != 0.


2



3, 12, 15, 17, 27, 30, 40, 42, 48, 60, 63, 68, 75, 77, 90, 95, 99, 105, 108, 112, 120, 130, 135, 140, 147, 153, 156, 160, 165, 168, 192, 195, 220, 240, 243, 252, 270, 272, 273, 300, 301, 308, 312, 315, 323, 350, 360, 363, 375, 378, 380, 396, 399, 420, 425, 432, 448, 462, 480, 495, 507
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OFFSET

1,1


COMMENTS

This sequence is closely related to A271576.
Conditions to be satisfied for a solution:
 z cannot be a square.
 z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
 If z has 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.
 If z is even, x and y must be even too.
 The lower limit of the ratio x/y is sqrt(2).
Multiple solutions are possible; e.g., term 420 has 5 solutions.


LINKS

Table of n, a(n) for n=1..61.


EXAMPLE

The 5 solutions corresponding to a(54) = 420 are 420^2 = 176400 = 444^2  12^4 = 580^2  20^4 = 609^2  21^4 = 1295^2  35^4 = 3164^2  56^4.


CROSSREFS

Cf. A000290, A000583, A271576, A002144, A002145.
Sequence in context: A227302 A201273 A221920 * A188549 A287357 A044828
Adjacent sequences: A349660 A349661 A349662 * A349664 A349665 A349666


KEYWORD

nonn


AUTHOR

KarlHeinz Hofmann and Hugo Pfoertner, Dec 09 2021


STATUS

approved



