A056813 Largest non-unitary prime factor of LCM(1,...,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset

1,4

Comments

For n>0, prime(n) appears {(prime(n+1))^2 - (prime(n))^2} times [from n=(prime(n))^2 to n=(prime(n+1))^2 - 1], that is, A000040(n) appears A069482(n) times (from n=A001248(n) to n=A084920(n+1)). - Lekraj Beedassy, Mar 31 2005

a(n) is the largest prime factor of A045948(n). [Matthew Vandermast, Oct 29 2008]

Alternative definition: a(n) = largest prime <= sqrt(n) (considering 1 as prime for this occasion, see A008578 for the 19th century definition of primes). - Jean-Christophe Hervé, Oct 29 2013

Links

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Formula

a(n) = prime(w) if prime(w)^2 <= n < prime(w+1)^2.

To get the sequence, repeat 1 three times, and then for any k >= 1, repeat A000040(k) A069482(k) times; or equivalently, for any k >= 1, repeat A008578(k) a number of times equal to A008578(k+1)^2 - A008578(k)^2. - Jean-Christophe Hervé, Oct 29 2013

Example

The j-th prime appears at the position of its square, at n = prime(j)^2.

Mathematica

Table[f = Transpose[FactorInteger[LCM @@ Range[n]]]; pos = Position[f[[2]], _?(# > 1 &)]; If[pos == {}, 1, f[[1, pos[[-1]]]][[1]]], {n, 100}] (* T. D. Noe, Oct 30 2013 *)

Crossrefs

Cf. A054041, A056170.

Cf. A000040, A001248, A008578, A069482.

Sequence in context: A105517 A069624 A230774 * A125269 A077430 A105513

Adjacent sequences: A056810 A056811 A056812 * A056814 A056815 A056816

Keyword

nonn,easy

Author

Labos Elemer, Aug 28 2000

Extensions

Corrected offset by Jean-Christophe Hervé, Oct 29 2013

Status

approved