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A178109 The maximum number d, 2 <= d <= n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.

%I A178109 #16 Jan 15 2019 16:19:43
%S A178109 0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,6,0,6,0,0,0,4,0,0,0,8,6,10,0,8,0,14,0,
%T A178109 14,0,15,0,14,9,14,0,9,0,10,15,20,0,22,0,22,21,18,0,21,20,22,24,24,0,
%U A178109 26,0,28,24,30,10,28,0,30,24,26,0,33,0,30,20,30,21,28,0,38,33,38,0,28,20,36
%N A178109 The maximum number d, 2 <= d <= n/2, which divides binomial(n-d-1,d-1) and is not coprime to n.
%C A178109 If no such divisors d exist, a(n)=0.
%H A178109 Robert Israel, <a href="/A178109/b178109.txt">Table of n, a(n) for n = 1..10000</a>
%H A178109 V. Shevelev, <a href="http://dx.doi.org/10.1142/S179304210700078X">On divisibility of binomial(n-i-1,i-1) by i</a>, Int. J. Number Theory vol 3, no. 1 (2007), 119-139.
%p A178109 A178109 := proc(n) local dvs,d ; dvs := {} ; for d from 1 to n/2 do if gcd(n,d) > 1 and d in numtheory[divisors]( binomial(n-d-1,d-1)) then dvs := dvs union {d} ; end if; end do:
%p A178109 if nops(dvs) = 0 then 0; else max(op(dvs)) ; end if; end proc:
%p A178109 seq(A178109(n),n=1..90) ; # _R. J. Mathar_, May 28 2010
%p A178109 # Alternative:
%p A178109 f:= proc(n) local d;
%p A178109   for d from floor(n/2) to 2 by -1 do
%p A178109      if igcd(d,n) > 1 and binomial(n-d-1,d-1) mod d = 0 then return d fi
%p A178109   od;
%p A178109   0
%p A178109 end proc:
%p A178109 map(f, [$1..100]); # _Robert Israel_, Jan 15 2019
%t A178109 a[n_] := If[n==1, 0, Module[{d=Floor[n/2]}, While[d>1 && (GCD[n, d]==1 || !Divisible[Binomial[n-d-1,d-1], d]), d--]; If[d==1, d=0]; d]]; Array[a, 100] (* _Amiram Eldar_, Dec 04 2018 *)
%Y A178109 Cf. A138389, A178071, A178098 - A178101, A178105
%K A178109 nonn,look
%O A178109 1,14
%A A178109 _Vladimir Shevelev_, May 20 2010
%E A178109 a(39), a(54) and a(70) corrected by _R. J. Mathar_, May 28 2010