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

Author

Labos Elemer, Aug 28 2000

Keywords

• nonn
• easy

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

Examples

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

Links

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

Crossrefs

Cf. A054041, A056170.

Cf. A000040, A001248, A008578, A069482.

Programs

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 *)

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

Extensions

Corrected offset by Jean-Christophe Hervé, Oct 29 2013