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A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

Sub-cases 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.

LINKS

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

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

Cf. A001358, A038456

Sequence in context: A124970 A251768 A338561 * A124886 A061195 A232111

Adjacent sequences:  A178031 A178032 A178033 * A178035 A178036 A178037

KEYWORD

nonn

AUTHOR

Michel Lagneau, May 17 2010

STATUS

approved

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Last modified November 23 17:26 EST 2020. Contains 338595 sequences. (Running on oeis4.)