A052180 Last filtering prime for n-th prime p: find smallest prime factor of each of the composite numbers between p and next prime; take maximal value.

2, 2, 3, 2, 3, 2, 3, 5, 2, 5, 3, 2, 3, 7, 5, 2, 5, 3, 2, 7, 3, 5, 7, 3, 2, 3, 2, 3, 11, 3, 7, 2, 11, 2, 5, 7, 3, 13, 5, 2, 11, 2, 3, 2, 11, 13, 3, 2, 3, 5, 2, 13, 11, 7, 5, 2, 5, 3, 2, 17, 13, 3, 2, 3, 17, 5, 11, 2, 3, 5, 19, 7, 13, 3, 5, 17, 3, 13, 7, 2, 7, 2, 19, 3, 5, 11, 3, 2, 3, 11, 13, 3, 17

Offset: 2

Author

Labos Elemer, Feb 05 2000

Keywords

• nonn
• easy
• nice

Comments

A000879(n) is the least index i such that a(i) = prime(n). - Labos Elemer, May 14 2003

Examples

			For n=9, n-th prime is 23, composites between 23 and next prime are 24 25 26 27 28, smallest prime divisors are 2 5 2 3 2; maximal value is 5, so a(9)=5.
		

Links

• T. D. Noe, Table of n, a(n) for n=2..10000

Crossrefs

Cf. A052248, A020639, A000720, A083269, A000879.

Cf. A010051.

Programs

Haskell

a052180 n = a052180_list !! (n-2)
a052180_list = f [4..] where
   f ws = (maximum $ map a020639 us) : f vs where
     (us, _:vs) = span  ((== 0) . a010051) ws
-- Reinhard Zumkeller, Dec 27 2012

Mathematica

ffi[x_] := Flatten[FactorInteger[x]];
lf[x_] := Length[FactorInteger[x]];
ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}];
mi[x_] := Min[ba[x]];
Table[Max[Table[mi[ba[w]], {w, Prime[j]+1, -1+Prime[j+1]}]], {j, 2, 256}]
(* Second program: *)
mpf[{a_,b_}] := Max[FactorInteger[#][[1,1]]& /@ Range[a+1,b-1]];
mpf/@ Partition[ Prime[Range[2,100]],2,1] (* Harvey P. Dale, Apr 30 2013 *)

PARI

a(n) = {my(p = prime(n), amax = 0); forcomposite(c = p, nextprime(p+1), amax = max(factor(c)[1,1], amax);); amax;} \\ Michel Marcus, Apr 21 2018

Python

from sympy import prime, nextprime, primefactors
def a(n):
  p = prime(n); q = nextprime(p)
  return max(min(primefactors(m)) for m in range(p+1, q))
print([a(n) for n in range(2, 95)]) # Michael S. Branicky, Feb 02 2021

Formula

a(n) = Max_{j=1+prime(n)..prime(n+1)-1} A020639(j) where A020639(j) is the least prime dividing j.