OFFSET
1,2
COMMENTS
In other words powerful numbers whose canonical prime factorization has no exponent that is a proper multiple of another exponent. - Peter Munn, Nov 04 2025
A number is powerful if its prime multiplicities are all greater than 1. Equivalently, if the number is divisible by the square of each of its prime divisors.
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 distinct and not pairwise indivisible, as 4 is a proper multiple of 2. Similar cases are any number with canonical prime factorization p^4*q^3*r^2, such as 10800, or p^4*q^5*r^2. Such numbers are not in the sequence.
36 = 2^2*3^2 is in the sequence as both exponents are 2, so neither exponent is a proper multiple of the other.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 01 2018
EXTENSIONS
Definition corrected and a(1)=1 inserted by Robert Israel, Jun 23 2019
STATUS
approved
