OFFSET
1,1
COMMENTS
Numbers whose only nontrivial square divisor is the square of their second smallest distinct prime factor. - Peter Munn, Aug 27 2026
Proper subset of A332785 (numbers neither squarefree nor powerful).
Superset of A095990 (numbers with unsorted prime signature (1,2)).
Terms have at least 2 distinct prime factors, therefore, rad(a(n)) is in A120944.
Unsorted prime signature of a(n) is (1,2), followed by zero or more 1's.
Smallest term with m distinct prime factors, with m >= 2, is 3*A002110(m).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
EXAMPLE
Table of n, a(n) for select n:
n a(n)
---------------------------
1 18 = 2 * 3^2
2 50 = 2 * 5^2
3 75 = 3 * 5^2
4 90 = 2 * 3^2 * 5
5 98 = 2 * 7^2
6 126 = 2 * 3^2 * 7
7 147 = 3 * 7^2
8 198 = 2 * 3^2 * 11
9 234 = 2 * 3^2 * 13
10 242 = 2 * 11^2
11 245 = 5 * 7^2
25 630 = 2 * 3^2 * 5 * 7
MATHEMATICA
nn = 1350; k = 3; mm = nn/k; Union@ Map[If[# <= nn, #, Nothing] &[#*FactorInteger[#][[2, 1]] ] &, Select[Range[mm], And[SquareFreeQ[#], CompositeQ[#] ] &] ]
PROG
(Python)
from math import prod
from sympy import factorint
def ok(k):
f = factorint(k)
lpf = min(f, default=0)
q = min(set(f)-{lpf}, default=0)
return q and k == q * prod(f)
print([k for k in range(1276) if ok(k)]) # Michael S. Branicky, Aug 02 2026
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Michael De Vlieger, Jul 30 2026
STATUS
approved