A212300 Median prime factor of 1..n.

2, 2, 2, 2, 2, 2, 3, 2, 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, 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, 11, 7, 7, 7, 11, 7, 11, 11, 11, 11, 11

Offset

1,1

Comments

Smallest prime p such that at least n/2 numbers up to n have no prime factors larger than p.

A246430(n) = smallest m such that a(m) = prime(n): a(A246430(n))=A000040(n). - Reinhard Zumkeller, Sep 01 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Naslund, The median largest prime factor and the mean of omega(n) (2012)

Formula

a(n) = k * n^(1/sqrt(e)) * (1 + O(1/log n)), where k = 0.7738... = A212299. See Nashlund 2012 for further references and a more precise result (Theorem 8).

Mathematica

a[1]=2; a[2]=2; a[n_] := NextPrime[Median[Table[FactorInteger[k][[-1, 1]], {k, 1, n}]] + 1/2, -1]; ( Charles R Greathouse IV, Jun 12 2013 )

Prog

(PARI) a(n)=my(v=vector(n, u, u)); forprime(p=2, n+1, forstep(i=p, n, p, v[i]/=p^valuation(i, p)); if(sum(i=1, n, v[i]==1)>=n/2, return(p))) \\ Charles R Greathouse IV, Jun 12 2013
(Haskell)
a212300 n = a212300_list !! (n-1)
a212300_list = f 1 (2 : a000040_list) [1] $ tail a006530_list where
   f x ps'@(_ : ps@(_ : p : _)) gpfs (q : qs) =
     y : f (x + 1) (if y == p then ps else ps') (q : gpfs) qs where
     y = head [z | z <- ps', length (filter (> z) gpfs) <= div x 2]
-- Reinhard Zumkeller, Sep 01 2014

Crossrefs

Cf. A212299.

Cf. A006530, A000040, A246430.

Sequence in context: A277879 A257071 A332025 * A165074 A374152 A273165

Adjacent sequences: A212297 A212298 A212299 * A212301 A212302 A212303

Keyword

nonn,nice

Author

Charles R Greathouse IV, Jul 03 2012

Status

approved