A391023 Numbers with more than 1 distinct prime factor and whose largest prime factor exponent is 2.

12, 18, 20, 28, 36, 44, 45, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 100, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 196, 198, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276

Offset

1,1

Comments

Disjoint union of A177492 and A386684.

A126706 is the disjoint union of this sequence and A378767.

Links

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

Example

Let s = A177492 and t = A386684.
Table of n, a(n) for select n:
  n  a(n)
----------------------------------
  1   12 = s(1) = 2^2 *  3
  2   18 = s(2) = 2   *  3^2
  3   20 = s(3) = 2^2 *  5
  4   28 = s(4) = 2^2 *  7
  5   36 = t(1) = 2^2 *  3^2
  6   44 = s(5) = 2^2 * 11
  7   45 = s(6) = 3^2 *  5
 10   60 = s(9) = 2^2 *  3   * 5
 20  100 = t(2) = 2^2 *  5^2
193  900 = t(8) = 2^2 *  3^2 * 5^2
.
4 is not in this sequence since 4 is a prime power.
24 = 2^3 * 3 is not in this sequence since the largest prime exponent exceeds 2.

Mathematica

Select[Range[300], And[Length[#] > 1, 1 < Max[#] <= 2] &[FactorInteger[#][[All, -1]] ] &]

Prog

(PARI) isok(k) = {my(f = factor(k)); omega(f) > 1 && vecmax(f[ , 2]) == 2; } \\ Amiram Eldar, Nov 29 2025
(Python)
from math import isqrt
from sympy import primepi, mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A391023(n):
    def f(x): return n+x+primepi(isqrt(x))+sum(mobius(k)*(x//k**j)*(-1 if j&1 else 1) for j in range(2, 4) for k in range(1, integer_nthroot(x, j)[0]+1))
    return bisection(f, n, n) # Chai Wah Wu, Jan 07 2026

Crossrefs

Cf. A004709, A067259, A126706, A177492, A378767, A386684.

Sequence in context: A162947 A393940 A351201 * A359892 A070011 A084679

Adjacent sequences: A391020 A391021 A391022 * A391024 A391025 A391026

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 27 2025

Status

approved