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

%I A122181 #17 Oct 05 2024 09:12:18
%S A122181 24,30,36,40,42,48,54,56,60,64,66,70,72,78,80,84,88,90,96,100,102,104,
%T A122181 105,108,110,112,114,120,126,128,130,132,135,136,138,140,144,150,152,
%U A122181 154,156,160,162,165,168,170,174,176,180,182,184,186,189,190,192,195
%N A122181 Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).
%C A122181 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).
%H A122181 Amiram Eldar, <a href="/A122181/b122181.txt">Table of n, a(n) for n = 1..10000</a>
%e A122181 a(1) = 24 = 2*3*4, a product of three distinct proper divisors (omega(24) = 2, bigomega(24) = 4).
%e A122181 a(2) = 30 = 2*3*5, a product of three distinct prime factors (omega(30) = 3).
%e A122181 a(10) = 64 = 2*4*8 [= 2^1*2^2*2^3] (omega(64) = 1, bigomega(64) = 6).
%t A122181 Select[Range[200], DivisorSigma[0, #] > 6 &] (* _Amiram Eldar_, Oct 05 2024 *)
%o A122181 (PARI) isok(n) = numdiv(n)>6
%o A122181 (PARI) isok(n) = (omega(n)==1 && bigomega(n)>5) || (omega(n)==2 && bigomega(n)>3) || (omega(n)>2)
%Y A122181 Cf. A122179, A122180, A000005, A070824, A000977, A001221, A001222, A001597, A076470.
%K A122181 nonn,easy
%O A122181 1,1
%A A122181 _Rick L. Shepherd_, Aug 24 2006