A338020 a(n) is the number of circles of positive integer area with radii less than n and greater than n - 1.

3, 9, 16, 22, 28, 35, 40, 48, 53, 60, 66, 72, 78, 85, 91, 98, 103, 110, 117, 122, 129, 135, 141, 148, 154, 160, 167, 173, 179, 185, 192, 197, 205, 210, 217, 223, 229, 236, 242, 248, 255, 260, 267, 274, 279, 286, 292, 299, 304, 311, 318, 323, 330, 336, 343, 349, 355, 361, 367

Offset: 1

Author

Torlach Rush, Oct 06 2020

Keywords

• nonn

Comments

Conjecture: k >= 2, each triple Tr(k) = {a(k), a(k+1), a(k+2)} gives the sides of an integer-sided triangle, and {(a(k+2) - a(k)), (a(k+2) - a(k+1)), (a(k+1) - a(k))} is a degenerate integer-sided triangle.

Crossrefs

Cf. A066643 (partial sums).

Programs

PARI

ap(n) = {my(x = 0, y = 1, ia = 1); while(y, if(n > sqrt(ia / Pi), x++; ia++, y = 0)); return(x)}
a(n) = {my(x = 0, y = 1, ia = 1); while(y, if(n > sqrt(ia / Pi), x++; ia++, y = 0)); return(x - ap(n-1))}
for(i = 1, 70, print1(a(i), ", "))

Formula

a(n) = #{floor(sqrt(k/Pi)) < n: n > 0, k > 0}.

a(n) = A066643(n)-A066643(n-1). - R. J. Mathar, Jan 25 2023