A203025 Largest perfect power divisor of n.

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1

Offset: 1

Author

Antonio Roldán, Dec 28 2011

Keywords

• nonn

Comments

This sequence shares many elements with A057521, but is not identical: A057521(72)=72 but a(72)=36.

Not multiplicative: a(49)=49; a(125)=125, a(49*125) = 1225 <> 49*125.

Examples

			a(40)=a(2^3*5)=2^3=8.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A057521, A087320, A274006.

Programs

Maple

A203025:=proc(n)
    local a,Le,d,i,k,pe;
    pe := ifactors(n)[2];
    Le := {seq(i[2],i=pe)} minus {1};
    a := 1;
    for k in Le do
        d := mul(i[1]^(k*floor(i[2]/k)), i=pe) ;
        a:=max(a,d);
    end do;
    a
end proc:
seq(A203025(n),n=1..10000); # Felix Huber, Jun 01 2025

Mathematica

Table[If[SquareFreeQ[n], 1, s = FactorInteger[n]; Max[Table[Times @@ Cases[s, {p_, ep_} :> p^i /; (ep >= i)], {i, 2, Max[s[[All, 2]]]}]]], {n, 100}] (* Olivier Gerard, Jun 03 2016 *)

PARI

a(n)=my(f=factor(n),mx=1);for(e=2,if(n>1,vecmax(f[,2])), mx=max(mx,prod(i=1,#f[,1],f[i,1]^(f[i,2]\e*e))));mx \\ Charles R Greathouse IV, Dec 28 2011

Formula

a(n) = max{ A001597(k) : A001597(k)|n }. - R. J. Mathar, Jun 09 2016

Extensions

Values matching definition restored by Franklin T. Adams-Watters, Jun 06 2016