A245803 Numerator of the partial sum of the number of prime factors function divided by n.

0, 1, 2, 1, 1, 7, 8, 11, 13, 3, 16, 19, 20, 11, 8, 7, 29, 16, 33, 9, 38, 20, 41, 15, 47, 49, 52, 55, 56, 59, 60, 65, 67, 69, 71, 25, 76, 39, 80, 21, 85, 44, 89, 23, 19, 97, 98, 103, 15, 54, 110, 113, 114, 59, 24, 31, 42, 64, 129, 133, 134, 68, 139, 145, 147, 25, 151, 77, 52, 159, 160, 55

Offset

1,3

Links

Table of n, a(n) for n=1..72.

Formula

a(n) is the numerator of (sum(i = 1 .. n, Omega(i)))/n, where Omega is the number of prime factors function (counted with multiplicity, see A001222).

Example

a(8) = 11, since the integers from 1 to 8 have between them a total of 11 prime factors, making our fraction here 11/8.
a(9) = 13, since the integers from 1 to 9 have between them a total of 13 prime factors, making our fraction here 13/9.
a(10) = 3, since the integers from 1 to 10 have between them a total of 15 prime factors, making our fraction here 15/10 = 3/2.

Maple

N:= 100: # to get terms a(1) to a(N)
map(numer, zip(`/`, ListTools:-PartialSums([seq(numtheory:-bigomega(n), n=1..N)]), [$1..N])); # Robert Israel, Aug 29 2014

Mathematica

Numerator[Array[Plus@@Last/@FactorInteger[#!] &, 120, 2]/Range[2, 121]]

Prog

(PARI) lista(nn) = {s = 0; for (n=1, nn, s += bigomega(n); print1(numerator(s/n), ", "); ); } \\ Michel Marcus, Aug 28 2014

Crossrefs

Cf. A022559.

Sequence in context: A120475 A086738 A351083 * A198941 A058955 A176055

Adjacent sequences: A245800 A245801 A245802 * A245804 A245805 A245806

Keyword

nonn,easy,frac

Author

Alonso del Arte, Aug 22 2014

Status

approved