A122181 Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).

24, 30, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195

Offset: 1

Author

Rick L. Shepherd, Aug 24 2006

Keywords

• nonn
• easy

Comments

Equivalently, numbers k with at least 7 divisors (A000005(k) > 6). Equivalently, numbers k with at least 5 proper divisors (A070824(k) > 4). Equivalently, numbers k such that i) k has at least three distinct prime factors (A000977), ii) k has two distinct prime factors and four or more total prime factors (k = p^j*q^m, p,q primes, j+m >= 4), or iii) k = p^m, a perfect power (A001597) but restricted to prime p and m >= 6 [= 1+2+3] (some terms of A076470).

Examples

			a(1) = 24 = 2*3*4, a product of three distinct proper divisors (omega(24) = 2, bigomega(24) = 4).
a(2) = 30 = 2*3*5, a product of three distinct prime factors (omega(30) = 3).
a(10) = 64 = 2*4*8 [= 2^1*2^2*2^3] (omega(64) = 1, bigomega(64) = 6).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A122179, A122180, A000005, A070824, A000977, A001221, A001222, A001597, A076470.

Programs

Mathematica

Select[Range[200], DivisorSigma[0, #] > 6 &] (* Amiram Eldar, Oct 05 2024 *)

PARI

isok(n) = numdiv(n)>6

PARI

isok(n) = (omega(n)==1 && bigomega(n)>5) || (omega(n)==2 && bigomega(n)>3) || (omega(n)>2)