A227797 Number of composites removed in each step in the Sieve of Eratosthenes for 10^8.

49999999, 16666666, 6666666, 3809523, 2077920, 1598400, 1128284, 950133, 743581, 564099, 509508, 413103, 362709, 337382, 301484, 261684, 230683, 219393, 196552, 182782, 175351, 159910, 150351, 138581, 125778, 119552, 116075, 110630, 107564, 102739, 90485

Offset

1,1

Comments

The number of composites <= 10^8 for which the n-th prime is the least prime factor.

pi(sqrt(10^8)) = the number of terms of A227797.

The sum of a(n) for n = 1..1229 = A000720(10^8) + A065855(10^8).

Links

Eric F. O'Brien, Table of n, a(n) for n = 1..1229

Formula

Writing floor(a/b) as [a / b]:

a(1) = [10^8 / 2] - 1.

a(2) = [10^8 / 3] - [10^8 / 6] - 1.

a(3) = [10^8 / 5] - [10^8 / 10] - [10^8 / 15] + [10^8 / 30] - 1.

a(4) = [10^8 / 7] - [10^8 / 14] - [10^8 / 21] - [10^8 / 35] + [10^8 / 42] + [10^8 / 70] + [10^8 / 105] - [10^8 / 210] - 1.

Example

For n = 3, prime(n) = 5, a(n) = 6666666: 5 divides 10^8 20000000 times. 10 is the least common multiple of 2 (prime(1)) and 5 and 15 is the least common multiple of 3 (prime(2)) and 5; thus [10^8 / 10] multiples of 5 and [10^8 / 15] multiples of 5 have already been eliminated by a(1) and a(2), and thereby respectively reduce a(3) by 10000000 and 6666666 offset by [10^8 / 30] multiples of 5 which would otherwise excessively reduce a(3) by 3333333 because 30 is the least common multiple of 2, 3 and 5. a(3) is further reduced by 1 as 5 itself is not eliminated.

Crossrefs

Cf. A133228, A145538-A145540, A227155, A227798, A227799, A145532-A145537.

Sequence in context: A017591 A271020 A358018 * A236947 A348069 A080796

Adjacent sequences: A227794 A227795 A227796 * A227798 A227799 A227800

Keyword

nonn,fini

Author

Eric F. O'Brien, Jul 31 2013

Status

approved