A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 864, 1024, 1152, 1280, 1536, 1728, 2048, 2304, 2560, 3072, 3456, 3840, 4096, 4608, 5120, 6144, 6912, 7680, 8192, 9216, 10240, 11520, 12288, 13824, 15360

Offset: 1

Author

Hugo Pfoertner, May 23 2002

Keywords

• nonn

Examples

			Circles with radius 1 and 2 have 4 lattice points on their circumference, so a(1)=4. A circle with radius sqrt(5) passes through 8 lattice points of the shape (2,1), so a(2)=8. A circle with radius 5 passes through 4 lattice points of shape (5,0) and through 8 points of shape (4,3), so a(3)=4+8=12
A071383(11) = 5^2 * 13^2 * 17^1 = 71825. Therefore A071385(11) = 4*(2+1)*(2+1)*(1+1) = 72.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..425
• Hugo Pfoertner, Construction of A071383, A071384, A071385

Crossrefs

Records in A004018. Cf. A071383, A071384, A002144, A002654, A300163, A344470.

Programs

PARI

my(v=list(10^15), rec=0); print1(1, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(4*rec, ", "))) \\ Jianing Song, May 20 2021, see program for A054994

Formula

a(n) = 4 * Product_{k=1..klim} (e_k + 1), where klim and e_1 >= e_2 >= ... >= e_klim > 0 are known from A071383(n) = Product_{k=1..klim} p_k^e_k, with p_k = k-th prime of the form 4i+1. (J. H. Conway)

a(n) = 4*A000005(A071383(n)) for n > 1.

a(n) = A004018(A071383(n)).

a(n) = A002654(A071383(n)) for n > 1. - Jianing Song, May 20 2021

a(n) = 4*A344470(n-1) for n > 1. - Hugo Pfoertner, Sep 04 2022