A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

1, 1, 1, 7, 1, 11, 1, 253, 17, 19, 1, 595, 1, 27, 29, 39711, 1, 1378, 1, 1711, 41, 43, 1, 138415, 49, 51, 3160, 3403, 1, 3916, 1, 25637001, 65, 67, 69, 477191, 1, 75, 77, 657359, 1, 7750, 1, 8515, 8911, 91, 1, 132563501, 97, 11026, 101, 11935, 1, 1633355

Offset: 1

Author

Michel Lagneau, May 17 2010

Keywords

• nonn

Comments

Omega(.) = A001222(.) is the number of prime divisors of n (counted with multiplicity).

binomial(nk,k)= n*binomial(nk-1,k-1) ensures that all entries are integers.

Subcases for this sequence:

If n is prime, Omega(n) = 1, and a(n) = binomial (n,1) / n = 1.

If n and n+1 are products of two primes (A070552), then Omega(n) = Omega(n+1) = 2, and binomial(n*Omega(n), Omega(n)) / n = binomial(2*n, 2) / n = 2*n-1 and binomial(2*(n+1), 2) / (n+1) = 2*n+1, and we obtain two consecutive numbers of the form (x, x+2), for example (17,19), (27,29), (41,43),... at n =9, 14...

Chaining this property: If n, n+1, and n+2 are semiprimes (A056809) , we find three consecutive numbers of the form (x, x+2,x+4), for example (65, 67, 69), (169, 171, 173), at n=33, 85.

At places where Omega(n)=3, we find the subsequence A060544, for example a(8) = A060544(8).

At places where Omega(n)=4, we find the subsequence A015219.

Examples

			a(8) = binomial(8*Omega(8),Omega(8))/8 = binomial(8*3,3)/8 = 2024/8 = 253.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001358, A038456

Programs

Maple

A178034 := proc(n)
        local o ;
        o := numtheory[bigomega](n) ;
        binomial(n*o,o)/n ;
end proc: # R. J. Mathar, Jul 08 2012

Mathematica

bon[n_]:=Module[{o=PrimeOmega[n]},Binomial[n*o,o]/n]; Array[bon,60] (* Harvey P. Dale, Jul 22 2014 *)

PARI

a(n)=my(b=bigomega(n));binomial(n*b,b)/n \\ Charles R Greathouse IV, Oct 25 2012