A317102 Powerful numbers whose distinct prime multiplicities are pairwise indivisible.

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 169, 196, 200, 216, 225, 243, 256, 288, 289, 343, 361, 392, 432, 441, 484, 500, 512, 529, 625, 648, 675, 676, 729, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Author

Gus Wiseman, Aug 01 2018

Keywords

• nonn

Comments

A number is powerful if its prime multiplicities are all greater than 1.

Examples

			144 = 2^4 * 3^2 is not in the sequence because 4 and 2 are not pairwise indivisible.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A001694, A056239, A112798, A118914, A124010, A285572, A285573, A303362, A304713, A316475, A317101, A317616.

Programs

Maple

filter:= proc(n) local L,i,j,q;
  L:= convert(map(t -> t[2],ifactors(n)[2]),set);
  if min(L) = 1 then return false fi;
  for j from 2 to nops(L) do
    for i from 1 to j-1 do
      q:= L[i]/L[j];
      if q::integer or (1/q)::integer then return false fi;
  od od;
  true
end proc:
select(filter, [$4..10000]); # Robert Israel, Jun 23 2019

Mathematica

Select[Range[1000],And[Max@@Last/@FactorInteger[#]>=2,Select[Tuples[Last/@FactorInteger[#],2],And[UnsameQ@@#,Divisible@@#]&]=={}]&]

Extensions

Definition corrected and a(1)=1 inserted by Robert Israel, Jun 23 2019