A328202 a(n) is the greatest common divisor of all the numbers in row n of Pascal's triangle excluding 1 and n.

6, 10, 5, 7, 14, 6, 3, 11, 11, 13, 13, 1, 2, 34, 17, 19, 19, 1, 1, 23, 23, 5, 5, 3, 3, 29, 29, 31, 62, 2, 1, 1, 1, 37, 37, 1, 1, 41, 41, 43, 43, 1, 1, 47, 47, 7, 7, 1, 1, 53, 53, 1, 1, 1, 1, 59, 59, 61, 61, 1, 2, 2, 1, 67, 67, 1, 1, 71, 71, 73, 73, 1, 1, 1, 1

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

Formula

a(n) = A014963(n) * A014963(n - 1) (conjecture). - Jon Maiga, Oct 08 2019

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

Cf. A007318 (Pascal's triangle), A014963.

Sequence in context: A144763 A280302 A083478 * A306368 A075368 A235117

Adjacent sequences: A328199 A328200 A328201 * A328203 A328204 A328205

Keyword

nonn,look

Author

Joel Kaufmann, Oct 07 2019

Extensions

More terms from Giovanni Resta, Oct 08 2019

Status

approved