%I #7 Jul 22 2024 15:24:21

%S 1,2,4,6,10,14,22,22,30,46,74,94,90,102,130,170,198,222,290,350,474,

%T 650,730,734,746,838,962,1214,2138,2582,1890,1830,2526,3498,4746,6842,

%U 5098,6358,8178,10634,8650,9782,13634,14438,17178,20202,22170,21422,16298

%N Sum of the numbers of unitary divisors of the binomial coefficients C(n,k), k=0..n.

%C Row sums of the triangle A103444.

%e a(3) = 6 because the divisors of 1,3,3,1 are {1},{1,3},{1,3},{1}, respectively, all of which are unitary, and 1 + 2 + 2 + 1 = 6.

%p 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 50 do b[n]:=[seq(T(n,k),k=0..n)] od: seq(sum(b[n][j],j=1..n+1),n=0..50);

%t a[n_] := Sum[2^PrimeNu[Binomial[n, k]], {k, 0, n}]; Array[a, 50, 0] (* _Amiram Eldar_, Jul 22 2024 *)

%o (PARI) a(n) = sum(k = 0, n, 2^omega(binomial(n, k))); \\ _Amiram Eldar_, Jul 22 2024

%Y Cf. A007318, A034444, A103444.

%K nonn

%O 0,2

%A _Emeric Deutsch_, Feb 06 2005