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

%I A073482 #38 Aug 29 2024 02:09:09
%S A073482 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,
%T A073482 7,43,23,47,17,53,11,19,29,59,61,31,13,11,67,23,7,71,73,37,11,13,79,
%U A073482 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
%N A073482 Largest prime factor of the n-th squarefree number.
%H A073482 Reinhard Zumkeller, <a href="/A073482/b073482.txt">Table of n, a(n) for n = 1..10000</a>
%H A073482 Jean-Marie De Koninck and Rafael Jakimczuk, <a href="https://doi.org/10.33044/revuma.3154">Summing the largest prime factor over integer sequences</a>, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.
%F A073482 a(n) = A006530(A005117(n)).
%F A073482 a(n) = A265668(n, A001221(n)). - _Reinhard Zumkeller_, Dec 13 2015
%F A073482 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
%p A073482 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
%t A073482 FactorInteger[#][[-1, 1]]& /@ Select[Range[100], SquareFreeQ] (* _Jean-François Alcover_, Feb 01 2018 *)
%t A073482 s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 2 &], f[[-1, 1]], Nothing]]; Array[s, 200] (* _Amiram Eldar_, Mar 03 2024 *)
%o A073482 (Haskell)
%o A073482 a073482 = a006530 . a005117  -- _Reinhard Zumkeller_, Feb 04 2012
%o A073482 (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
%o A073482 (Python)
%o A073482 from math import isqrt
%o A073482 from sympy import mobius, primefactors
%o A073482 def A073482(n):
%o A073482     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A073482     kmin, kmax = 0,1
%o A073482     while f(kmax) > kmax:
%o A073482         kmax <<= 1
%o A073482     while kmax-kmin > 1:
%o A073482         kmid = kmax+kmin>>1
%o A073482         if f(kmid) <= kmid:
%o A073482             kmax = kmid
%o A073482         else:
%o A073482             kmin = kmid
%o A073482     return max(primefactors(kmax),default=1) # _Chai Wah Wu_, Aug 28 2024
%Y A073482 Cf. A073481, A001221, A005117, A006530, A265668, A323669.
%K A073482 nonn,easy
%O A073482 1,2
%A A073482 _Reinhard Zumkeller_, Aug 03 2002
%E A073482 More terms from _Jason Earls_, Aug 06 2002