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

1,14

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.

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 *)

Crossrefs

Cf. A138389, A178071, A178098 - A178101, A178105

Sequence in context: A076290 A198224 A178105 * A339873 A055672 A327964

Adjacent sequences: A178106 A178107 A178108 * A178110 A178111 A178112

Keyword

nonn,look

Author

Vladimir Shevelev, May 20 2010

Extensions

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

Status

approved