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

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, 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, 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

Offset: 1

Author

Vladimir Shevelev, May 20 2010

Keywords

• nonn
• look

Comments

If no such divisors d exist, a(n)=0.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Int. J. Number Theory vol 3, no. 1 (2007), 119-139.

Crossrefs

Cf. A138389, A178071, A178098 - A178101, A178105

Programs

Maple

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:
if nops(dvs) = 0 then 0; else max(op(dvs)) ; end if; end proc:
seq(A178109(n),n=1..90) ; # R. J. Mathar, May 28 2010
# Alternative:
f:= proc(n) local d;
  for d from floor(n/2) to 2 by -1 do
     if igcd(d,n) > 1 and binomial(n-d-1,d-1) mod d = 0 then return d fi
  od;
  0
end proc:
map(f, [$1..100]); # Robert Israel, Jan 15 2019

Mathematica

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 *)

Extensions

a(39), a(54) and a(70) corrected by R. J. Mathar, May 28 2010