

A071383


Squared radii of the circles around (0,0) that contain record numbers of lattice points.


15



0, 1, 5, 25, 65, 325, 1105, 4225, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125
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OFFSET

1,3


COMMENTS

The number of lattice points (i,j) on the circle with i^2 + j^2 = a(n) is given by A071385(n).
In a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points", James R. Buddenhagen asked: Which circles have the property that they pass through more lattice points than any smaller circle? and he gave the terms 1, 25, 65, 325, 1105, 4225, 5525, with the missing 5 added by Ahmed Fares. In the same thread Gerry Myerson mentioned the factorization into primes of the form 4*k+1.
Also, numbers with a record number of divisors all of whose prime factors are of the form 4k + 1.  Amiram Eldar, Sep 12 2019


LINKS

Ray Chandler, Table of n, a(n) for n = 1..425 (first 97 terms from Ray Chandler, terms 98365 from Amiram Eldar)
James Buddenhagen, Circle with 3 lattice points, thread in sci.math (May 2002)
Hugo Pfoertner, Construction of the sequences A071383, A071384, A071385


FORMULA

For n>1 we have 1 < a(n+1)/a(n) <= 5, since one can multiply the points x+iy for which x^2 + y^2 = N by either 2+i or 2i to get two new sets of points X+iY for which X^2 + Y^2 = 5N. This strictly increases the number since it is easy to see that the two sets aren't the same.  J. H. Conway, Jun 04 2002
lim n >infinity Log(a(n))/n = 1. [Conjectured by Benoit Cloitre, proved by J. H. Conway]
Numbers of the form 5^e1*13^e2*17^e3 .. *pk^ek where pk is the kth prime of the form 4*n+1 with e1>=e2>=e3>=..>=ek.


CROSSREFS

Cf. A000448, A048610, A052199, A071384, A071385, A230655, A300162. Subsequence of A054994 (excluding first term). Where records occur in A004018.
Sequence in context: A054994 A108403 A007058 * A088959 A018782 A146665
Adjacent sequences: A071380 A071381 A071382 * A071384 A071385 A071386


KEYWORD

nonn


AUTHOR

Hugo Pfoertner, May 23 2002


STATUS

approved



