A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

1, 1, 1, 2, 3, 1, 8, 14, 7, 1, 48, 92, 56, 13, 1, 480, 968, 652, 186, 23, 1, 5760, 12096, 8792, 2884, 462, 35, 1, 92160, 199296, 152768, 54936, 10276, 1022, 51, 1, 1658880, 3679488, 2949120, 1141616, 239904, 28672, 1940, 69, 1, 36495360, 82607616

Offset: 0

Author

Matthew Vandermast, Jun 24 2004

Keywords

• nonn
• tabl

Comments

Sum of entries in n-th row is A002110(n), the product of the first n primes (primorial numbers, first definition).

From Peter Munn, Apr 10 2017: (Start)

T(n,k) is a count of those integers in any interval of A002110(n) integers that have exactly k of the first n primes as divisors. The count is the same for each such interval because each of the first n primes is a factor of an integer m if and only if it is a factor of m + A002110(n).

A284411(m) is least p=prime(n) such that 2*Sum_{k=0..m-1} T(n,k) < A002110(n).

(End)

Examples

			Triangle begins:
1
1 1
2 3 1
8 14 7 1
48 92 56 13 1
480 968 652 186 23 1
		

Crossrefs

First column is A005867; second column is A078456. See also A096180.

Cf. A194156, A284411.

Programs

PARI

primo(n) = prod(k=1, n, prime(k));
row(n) = {v = vector(n+1); for (k=1, primo(n), f = factor(k)[,1]; v[1+sum(j=1, #f, primepi(f[j])<=n)]++;); v;} \\ Michel Marcus, Apr 29 2017