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A081210 In prime factorization of n replace each prime power p^e with the greatest squarefree number <= p^e.

%I A081210 #27 Jun 09 2025 09:02:56
%S A081210 1,2,3,3,5,6,7,7,7,10,11,9,13,14,15,15,17,14,19,15,21,22,23,21,23,26,
%T A081210 26,21,29,30,31,31,33,34,35,21,37,38,39,35,41,42,43,33,35,46,47,45,47,
%U A081210 46,51,39,53,52,55,49,57,58,59,45,61,62,49,62,65,66,67,51,69,70,71,49,73
%N A081210 In prime factorization of n replace each prime power p^e with the greatest squarefree number <= p^e.
%H A081210 Reinhard Zumkeller, <a href="/A081210/b081210.txt">Table of n, a(n) for n = 1..10000</a>
%F A081210 Multiplicative with a(p^e) = A070321(p^e), p prime.
%F A081210 a(n) <= n and a(n) = n iff n is squarefree (A005117).
%F A081210 A081211(n) = a(a(n)), see A081212, A081213 and A081214 for iterations until a fixed point is reached.
%p A081210 A081210 := proc(n)
%p A081210     local a,pe;
%p A081210     a :=1 ;
%p A081210     for pe in ifactors(n)[2] do
%p A081210         a := a*A070321(op(1,pe)^op(2,pe)) ;
%p A081210     end do:
%p A081210     a ;
%p A081210 end proc:
%p A081210 seq(A081210(n),n=1..100) ; # _R. J. Mathar_, May 25 2023
%t A081210 gsf[n_] := For[k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; a[n_] := Times @@ gsf /@ Power @@@ FactorInteger[n]; Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Mar 27 2013 *)
%o A081210 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1], my(k = f[i,1]^f[i,2]); while(!issquarefree(k), k--); k));} \\ _Amiram Eldar_, Jun 09 2025
%Y A081210 Cf. A005117, A070321, A081212, A081213, A081214.
%K A081210 nonn,mult
%O A081210 1,2
%A A081210 _Reinhard Zumkeller_, Mar 10 2003