A073482 Largest prime factor of the n-th squarefree number.

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

Author

Reinhard Zumkeller, Aug 03 2002

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A073481, A001221, A005117, A006530, A265668, A323669.

Programs

Haskell

a073482 = a006530 . a005117  -- Reinhard Zumkeller, Feb 04 2012

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 *)

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

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

Extensions

More terms from Jason Earls, Aug 06 2002