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

2,1

Comments

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

Links

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

Formula

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A010051.

Sequence in context: A308050 A248147 A087458 * A065151 A345874 A338720

Adjacent sequences: A052177 A052178 A052179 * A052181 A052182 A052183

Keyword

nonn,easy,nice

Author

Labos Elemer, Feb 05 2000

Status

approved