A103444 Triangle read by rows: T(n,k) is number of unitary divisors of C(n,k), 0<=k<=n.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 4, 4, 4, 4, 4, 1, 1, 2, 4, 4, 4, 4, 2, 1, 1, 2, 4, 4, 8, 4, 4, 2, 1, 1, 2, 4, 8, 8, 8, 8, 4, 2, 1, 1, 4, 4, 8, 16, 8, 16, 8, 4, 4, 1, 1, 2, 4, 8, 16, 16, 16, 16, 8, 4, 2, 1, 1, 4, 8, 8, 8, 8, 16, 8, 8, 8, 8, 4, 1, 1, 2, 8, 8, 8, 8, 16, 16, 8, 8

Offset

0,5

Comments

Row n contains n+1 terms. Row sums yield A103445.

Links

Table of n, a(n) for n=0..100.

Example

T(6,3) = 4 because the divisors of C(6,3) = 20 are 1,2,4,5,10,20 of which 1,4,5,20 are unitary (i.e. d|20 such that gcd(d,20/d) = 1).
Triangle begins:
  1;
  1,1;
  1,2,1;
  1,2,2,1;
  1,2,4,2,1;
  1,2,4,4,2,1;

Maple

with(numtheory):unitdiv:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k], n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end: T:=proc(n, k) if k<=n then nops(unitdiv(binomial(n, k))) else 0 fi end: for n from 0 to 13 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

Table[2^PrimeNu[Binomial[n, k]], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 22 2024 *)

Prog

(PARI) T(n, k) = 2^omega(binomial(n, k)); \\ Amiram Eldar, Jul 22 2024

Crossrefs

Cf. A007318, A034444, A103445.

Sequence in context: A144464 A138015 A327742 * A099172 A214246 A214257

Adjacent sequences: A103441 A103442 A103443 * A103445 A103446 A103447

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 06 2005

Status

approved