A096294 - OEIS

%I #15 Jul 29 2019 13:13:13

%S 1,1,1,2,3,1,8,14,7,1,48,92,56,13,1,480,968,652,186,23,1,5760,12096,

%T 8792,2884,462,35,1,92160,199296,152768,54936,10276,1022,51,1,1658880,

%U 3679488,2949120,1141616,239904,28672,1940,69,1,36495360,82607616

%N 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).

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

%C From _Peter Munn_, Apr 10 2017: (Start)

%C 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).

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

%C (End)

%e Triangle begins:

%e 1

%e 1 1

%e 2 3 1

%e 8 14 7 1

%e 48 92 56 13 1

%e 480 968 652 186 23 1

%o (PARI) primo(n) = prod(k=1, n, prime(k));

%o 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

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

%Y Cf. A194156, A284411.

%K nonn,tabl

%O 0,4

%A _Matthew Vandermast_, Jun 24 2004