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A116921 a(n) = largest integer <= n/2 which is coprime to n.

%I A116921 #39 Jul 31 2024 13:14:45
%S A116921 0,1,1,1,2,1,3,3,4,3,5,5,6,5,7,7,8,7,9,9,10,9,11,11,12,11,13,13,14,13,
%T A116921 15,15,16,15,17,17,18,17,19,19,20,19,21,21,22,21,23,23,24,23,25,25,26,
%U A116921 25,27,27,28,27,29,29,30,29,31,31,32,31,33,33,34,33,35,35,36,35,37,37
%N A116921 a(n) = largest integer <= n/2 which is coprime to n.
%C A116921 a(n) + A116922(n) = n. For n>= 3, A116922(n) - a(n) is 1 if n is odd, is 2 if n is a multiple of 4 and is 4 if n is congruent to 2 (mod 4).
%C A116921 The arithmetic function v+-(n,2) as defined in A290988. - _Robert Price_, Aug 22 2017
%H A116921 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A116921 For n >= 3, a(n) = (n-1)/2 if n is odd, a(n) = n/2 - 1 if n is a multiple of 4 and a(n) = n/2 - 2 if n is congruent to 2 (mod 4).
%F A116921 a(n) = (2*n-4-2*(-1)^n+(-1)^(n/2)+(-1)^(3*n/2))/4, n>2. - _Wesley Ivan Hurt_, May 26 2015
%F A116921 For n > 2, a(n) = (n-2+cos(n*Pi/2)-cos(n*Pi))/2. - _Wesley Ivan Hurt_, Oct 02 2017
%F A116921 G.f.: t^2*(1+t^3-2*t^4+2*t^5)/((1-t)*(1-t^4)). - _Mamuka Jibladze_, Aug 22 2019
%t A116921 Join[{0, 1}, Table[(2 n - 4 - 2 (-1)^n + (-1)^(n/2) + (-1)^(3 n/2))/4, {n, 3, 50}]] (* _Wesley Ivan Hurt_, May 26 2015 *)
%t A116921 Table[Which[OddQ[n],(n-1)/2,Divisible[n,4],n/2-1,Mod[n,4]==2,n/2-2],{n,80}]//Abs (* _Harvey P. Dale_, Jun 24 2017 *)
%o A116921 (Magma) [0] cat [(2*n-4-2*(-1)^n+(-1)^(n div 2)+(-1)^(3*n div 2)) div 4: n in [3..80]]; // _Vincenzo Librandi_, May 26 2015
%o A116921 (PARI) a(n) = {forstep(k = n\2, 0, -1, if (gcd(n, k) == 1, return (k)););} \\ _Michel Marcus_, May 26 2015
%o A116921 (PARI) a(n) = {if(n%2, (n-1)/2, if(n==2, 1, n/2 - if(n%4, 2, 1)))} \\ _Andrew Howroyd_, Aug 22 2019
%o A116921 (Python)
%o A116921 def A116921(n): return n>>1 if n&1 or n==2 else (n>>1)-(2 if n&2 else 1) # _Chai Wah Wu_, Jul 31 2024
%Y A116921 Cf. A116922, A290988.
%K A116921 easy,nonn
%O A116921 1,5
%A A116921 _Leroy Quet_, Feb 26 2006
%E A116921 More terms from Wyatt Lloyd (wal118(AT)psu.edu), Mar 25 2006