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Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y) > 1}.
7

%I #25 Mar 29 2024 09:14:23

%S 0,1,2,5,6,13,14,21,26,37,38,53,54,69,82,97,98,121,122,145,162,185,

%T 186,217,226,253,270,301,302,345,346,377,402,437,458,505,506,545,574,

%U 621,622,681,682,729,770,817,818,881,894,953,990,1045,1046,1117,1146,1209

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

%H Reinhard Zumkeller, <a href="/A100613/b100613.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A000290(n) - A018805(n) = A185670(n) + A063985(n). - _Reinhard Zumkeller_, Jan 21 2013

%F a(n) = Sum_{k = 2..n} A242114(n,k). - _Reinhard Zumkeller_, May 04 2014

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

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

%o (Haskell)

%o a100613 n = length [()| x <- [1..n], y <- [1..n], gcd x y > 1]

%o -- _Reinhard Zumkeller_, Jan 21 2013

%o (PARI) a(n) = sum(i=1, n, sum(j=1, n, gcd(i,j)>1)); \\ _Michel Marcus_, Jan 30 2017

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A100613(n): # based on second formula in A018805

%o if n == 0:

%o return 0

%o c, j = 1, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c += (j2-j)*(k1**2-A100613(k1))

%o j, k1 = j2, n//j2

%o return n+c-j # _Chai Wah Wu_, Mar 24 2021

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

%K nonn

%O 1,3

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