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A073482
Largest prime factor of the n-th squarefree number.
13
1, 2, 3, 5, 3, 7, 5, 11, 13, 7, 5, 17, 19, 7, 11, 23, 13, 29, 5, 31, 11, 17, 7, 37, 19, 13, 41, 7, 43, 23, 47, 17, 53, 11, 19, 29, 59, 61, 31, 13, 11, 67, 23, 7, 71, 73, 37, 11, 13, 79, 41, 83, 17, 43, 29, 89, 13, 31, 47, 19, 97, 101, 17, 103, 7, 53, 107, 109, 11, 37, 113, 19, 23, 59, 17, 61, 41
OFFSET
1,2
LINKS
Jean-Marie De Koninck and Rafael Jakimczuk, Summing the largest prime factor over integer sequences, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.
FORMULA
a(n) = A006530(A005117(n)).
a(n) = A265668(n, A001221(n)). - Reinhard Zumkeller, Dec 13 2015
Sum_{A005117(n) <= x} a(n) = Sum_{i=1..k} d_i * x^2/log(x)^i + O(x^2/log(x)^(k+1)), for any given positive integer k, where d_i are constants, d_1 = 15/(2*Pi^2) = 0.759908... (A323669) (De Koninck and Jakimczuk, 2024). - Amiram Eldar, Mar 03 2024
MAPLE
issquarefree := proc(n::integer) local nf, ifa, lar; nf := op(2, ifactors(n)); for ifa from 1 to nops(nf) do lar := op(1, op(ifa, nf)); if op(2, op(ifa, nf)) >= 2 then RETURN(0); fi; od : RETURN(lar); end: printf("1, "); for n from 2 to 100 do lfa := issquarefree(n); if lfa > 0 then printf("%a, ", lfa); fi; od : # R. J. Mathar, Apr 02 2006
MATHEMATICA
FactorInteger[#][[-1, 1]]& /@ Select[Range[100], SquareFreeQ] (* Jean-François Alcover, Feb 01 2018 *)
s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 2 &], f[[-1, 1]], Nothing]]; Array[s, 200] (* Amiram Eldar, Mar 03 2024 *)
PROG
(Haskell)
a073482 = a006530 . a005117 -- Reinhard Zumkeller, Feb 04 2012
(PARI) do(x)=my(v=List([1])); forfactored(n=2, x\1, if(vecmax(n[2][, 2])==1, listput(v, vecmax(n[2][, 1])))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017
(Python)
from math import isqrt
from sympy import mobius, primefactors
def A073482(n):
def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
kmin, kmax = 0, 1
while f(kmax) > kmax:
kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return max(primefactors(kmax), default=1) # Chai Wah Wu, Aug 28 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Aug 03 2002
EXTENSIONS
More terms from Jason Earls, Aug 06 2002
STATUS
approved