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

1,2

Comments

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

Links

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

Example

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

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@@#]&]=={}]&]

Crossrefs

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

Sequence in context: A348121 A080366 A001694 * A157985 A001597 A359493

Adjacent sequences: A317099 A317100 A317101 * A317103 A317104 A317105

Keyword

nonn

Author

Gus Wiseman, Aug 01 2018

Extensions

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

Status

approved