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%I #33 Sep 08 2022 08:45:12
%S 0,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,2,0,2,1,1,0,3,0,1,0,2,0,6,0,0,1,1,
%T 1,4,0,1,1,3,0,6,0,2,2,1,0,4,0,2,1,2,0,3,1,3,1,1,0,11,0,1,2,0,1,6,0,2,
%U 1,6,0,6,0,1,2,2,1,6,0,4,0,1,0,11,1,1,1,3,0,11,1,2,1,1,1,5,0,2,2,4,0,6,0,3,6
%N Number of coprime pairs of divisors > 1 of n.
%C Also the number of divisors of n^2 which do not divide n and which are less than n. See link for proof. - _Andrew Weimholt_, Dec 06 2009
%C a(A000961(n)) = 0; a(A006881(n)) = 1; a(A054753(n)) = 2; a(A065036(n)) = 3. - _Robert G. Wilson v_, Dec 16 2009
%C First occurrence of k beginning with 0: 1, 6, 12, 24, 36, 96, 30, 384, 144, 216, 288, 60, 432, 24576, 1152, 864, 120, 393216, 1728, 1572864, 180, 240, 18432, 25165824, 5184, 210, 480, 13824, 10368, 360, 15552, 960, 20736, 55296, 1179648, 31104, 900, ..., . Except for 1, each is divisible by 6. Also the first occurrence of k must occur at or before 6*2^(n-1). - _Robert G. Wilson v_, Dec 16 2009
%C a(3*2^n) = n; if x = 2^n, then a(x) = a(2*x); and if x is not a power of two, then a(x) = y then a(2*x) > y. - _Robert G. Wilson v_, Dec 16 2009
%C a(n) = 0 iff n is a prime power. - _Franklin T. Adams-Watters_, Aug 20 2013
%H Reinhard Zumkeller, <a href="/A089233/b089233.txt">Table of n, a(n) for n = 1..10000</a>
%H Andrew Weimholt, <a href="/A089233/a089233.txt">Proof of an alternative characterization</a>
%F a(n) = #{(x,y): 1 < x < y, x|n, y|n and gcd(x, y) = 1}.
%F a(n) = A063647(n) - A000005(n) + 1.
%F a(n) = A018892(n) - A000005(n). - _Franklin T. Adams-Watters_, Aug 20 2013
%t a[n_] := (DivisorSigma[0, n^2] - 1)/2 - DivisorSigma[0, n] + 1; Array[a, 104] (* _Robert G. Wilson v_, Dec 16 2009 *)
%o (Haskell)
%o a089233 n = sum $ [a063524 $ gcd u v | let ds = tail $ a027750_row n,
%o u <- ds, v <- dropWhile (<= u) ds]
%o -- _Reinhard Zumkeller_, Sep 04 2013
%o (PARI) a(n) = (numdiv(n^2)-1)/2 - numdiv(n) + 1; \\ _Michel Marcus_, Feb 17 2016
%o (Magma) [(NumberOfDivisors(n^2)-1)/2 - NumberOfDivisors(n)+1: n in [1..100]]; // _Vincenzo Librandi_, Dec 23 2018
%K nonn
%O 1,12
%A _Reinhard Zumkeller_, Dec 11 2003
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