A100613 Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y) > 1}.

0, 1, 2, 5, 6, 13, 14, 21, 26, 37, 38, 53, 54, 69, 82, 97, 98, 121, 122, 145, 162, 185, 186, 217, 226, 253, 270, 301, 302, 345, 346, 377, 402, 437, 458, 505, 506, 545, 574, 621, 622, 681, 682, 729, 770, 817, 818, 881, 894, 953, 990, 1045, 1046, 1117, 1146, 1209

Offset: 1

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 02 2004

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A018805, A000290, A185670, A063985, A242114, A229099.

Programs

Haskell

a100613 n = length [()| x <- [1..n], y <- [1..n], gcd x y > 1]
-- Reinhard Zumkeller, Jan 21 2013

Mathematica

f[n_] := Table[ #^2 &[m], {m, 1, n + 1}] - FoldList[Plus, 1, 2 Array[EulerPhi, n, 2]] (* Gregg K. Whisler, Jun 25 2008 *)

PARI

a(n) = sum(i=1, n, sum(j=1, n, gcd(i,j)>1)); \\ Michel Marcus, Jan 30 2017

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A100613(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(k1**2-A100613(k1))
        j, k1 = j2, n//j2
    return n+c-j # Chai Wah Wu, Mar 24 2021

Formula

a(n) = A000290(n) - A018805(n) = A185670(n) + A063985(n). - Reinhard Zumkeller, Jan 21 2013

a(n) = Sum_{k = 2..n} A242114(n,k). - Reinhard Zumkeller, May 04 2014

a(n) ~ kn^2, where k = 1 - 6/Pi^2 = 0.39207... (A229099). - Charles R Greathouse IV, Mar 29 2024