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

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, 26, 21, 29, 30, 31, 31, 33, 34, 35, 21, 37, 38, 39, 35, 41, 42, 43, 33, 35, 46, 47, 45, 47, 46, 51, 39, 53, 52, 55, 49, 57, 58, 59, 45, 61, 62, 49, 62, 65, 66, 67, 51, 69, 70, 71, 49, 73

Offset: 1

Author

Reinhard Zumkeller, Mar 10 2003

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A005117, A070321, A081212, A081213, A081214.

Programs

Maple

A081210 := proc(n)
    local a,pe;
    a :=1 ;
    for pe in ifactors(n)[2] do
        a := a*A070321(op(1,pe)^op(2,pe)) ;
    end do:
    a ;
end proc:
seq(A081210(n),n=1..100) ; # R. J. Mathar, May 25 2023

Mathematica

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 *)

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

Formula

Multiplicative with a(p^e) = A070321(p^e), p prime.

a(n) <= n and a(n) = n iff n is squarefree (A005117).

A081211(n) = a(a(n)), see A081212, A081213 and A081214 for iterations until a fixed point is reached.