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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.
24
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
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
KEYWORD
nonn,easy,nice
AUTHOR
Labos Elemer, Feb 05 2000
STATUS
approved