A398802 - OEIS

A398802

Numbers k of the form q(k)^2 * rad(k), where rad = A007947, and q(k) the second smallest distinct prime factor of k.

54, 250, 270, 375, 378, 594, 686, 702, 918, 1026, 1029, 1242, 1566, 1674, 1715, 1750, 1890, 1998, 2214, 2322, 2538, 2625, 2662, 2750, 2862, 2970, 3186, 3250, 3294, 3510, 3618, 3834, 3942, 3993, 4125, 4158, 4250, 4266, 4394, 4482, 4590, 4750, 4806, 4875, 4914, 5130

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OFFSET

1,1

COMMENTS

Terms have at least 2 distinct prime factors.

Squarefree kernels are in A120944.

Unsorted prime signature {1,3} followed by zero or more 1's.

Superset of A397487.

Smallest term with j > 1 distinct prime factors is 3^2 * A002110(j).

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

EXAMPLE

Table of n, a(n) for select n:

n a(n)

-----------------------------------------

1 54 = 2 * 3^3

2 250 = 2 * 5^3

3 270 = 2 * 3^3 * 5

4 375 = 3 * 5^3

5 378 = 2 * 3^3 * 7

6 594 = 2 * 3^3 * 11

7 686 = 2 * 7^3

8 702 = 2 * 3^3 * 13

9 918 = 2 * 3^3 * 17

17 1890 = 2 * 3^3 * 5 * 7

179 20790 = 2 * 3^3 * 5 * 7 * 11

2277 270270 = 2 * 3^3 * 5 * 7 * 11 * 13

MATHEMATICA

nn = 5200; Set[{s, t}, {{2}, {2}}]; om = Max[s]; Union@ Map[If[# <= nn, #, Nothing] &[ #*Inner[#1^#2 &, FactorInteger[#][[s, 1]], t, Times] ] &, Select[Select[Range[2, nn], SquareFreeQ], PrimeNu[#] >= om &] ]

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**2 * prod(f)

print([k for k in range(5131) if ok(k)]) # Michael S. Branicky, Aug 13 2026

CROSSREFS

Cf. A005117, A007947, A119288, A120944, A126706, A332785, A397487.