OFFSET
4,1
COMMENTS
If p is prime, a(p) is a multiple of p and no term a(n) where n < p is a multiple of p.
From Robert Israel, Oct 16 2019: (Start)
If p is prime, a(p^k) and a(p^k+1) are divisible by p for all k>=1.
If p is a Fermat prime > 3, a(p)=2*p.
If p is a Mersenne prime, a(p+1)=2*p.
Conjecture: these are the only cases where a(n)>n. (End)
LINKS
Robert Israel, Table of n, a(n) for n = 4..10000
EXAMPLE
For n=8, take row 8 of Pascal's triangle:
1 8 28 56 70 56 28 8 1,
remove the first and last 2 numbers:
28 56 70 56 28,
the greatest common divisor of 28, 56, 70, 56, 28 is 14, thus a(8)=14.
MAPLE
f:= proc(n) local k, g;
g:= binomial(n, 2);
for k from 3 to n/2 do
g:= igcd(g, binomial(n, k));
if g = 1 then return g fi;
od;
g
end proc:
map(f, [$4..100]); # Robert Israel, Oct 16 2019
MATHEMATICA
a[n_] := GCD @@ Binomial[n, Range[2, n/2]]; a /@ Range[4, 90] (* Giovanni Resta, Oct 08 2019 *)
PROG
(PARI) a(n) = gcd(vector((n+1)\2-1, k, binomial(n, k+1))); \\ Michel Marcus, Oct 08 2019
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Joel Kaufmann, Oct 07 2019
EXTENSIONS
More terms from Giovanni Resta, Oct 08 2019
STATUS
approved