A102467 Positive integers k such that d(k) <> Omega(k) + omega(k), where d = A000005, Omega = A001222 and omega = A001221.

1, 12, 18, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156

Offset: 1

Author

Reinhard Zumkeller, Jan 09 2005

Keywords

• nonn

Comments

These are the numbers which are neither prime powers (>1) nor semiprimes. - M. F. Hasler, Jan 31 2008

For n > 1, positive integers k with a composite divisor, d < k, that is relatively prime to k/d. For example 12 is in the sequence since 4 (composite) is coprime to 12/4 = 3. - Wesley Ivan Hurt, Apr 25 2020

Examples

			10 is not in the sequence since d(10) = 4 is equal to Omega(10) + omega(10) = 2 + 2 = 4.
12 is in the sequence since d(12) = 6 is not equal to Omega(12) + omega(12) = 3 + 2 = 5. - _Wesley Ivan Hurt_, Apr 25 2020
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Joerg Arndt, Matters Computational (The Fxtbook), section 40.4.1.3 "Testing for irreducibility without GCD computations", pp. 839-840.

Crossrefs

Cf. A000005 (tau), A001221 (omega), A001222 (Omega).

Cf. A102466, A135767.

Programs

Haskell

a102467 n = a102467_list !! (n-1)
a102467_list = [x | x <- [1..], a000005 x /= a001221 x + a001222 x]
-- Reinhard Zumkeller, Dec 14 2012

Maple

with(numtheory):
q:= n-> is(tau(n)<>bigomega(n)+nops(factorset(n))):
select(q, [$1..200])[];  # Alois P. Heinz, Jul 14 2023

Mathematica

Select[Range[200], DivisorSigma[0, #] != PrimeOmega[#] + PrimeNu[#]&] (* Jean-François Alcover, Jun 22 2018 *)

PARI

is(n)=my(f=factor(n)[,2]); #f!=1 && f!=[1,1]~ \\ Charles R Greathouse IV, Oct 19 2015

Sage

def is_A102467(n) :
    return sloane.A001221(n) != 1 and sloane.A001222(n) != 2
def A102467_list(n) :
    return [k for k in (1..n) if is_A102467(k)]
A102467_list(156)  # Peter Luschny, Feb 07 2012

Formula

Complement of A102466; A000005(a(n)) <> A001221(a(n)) + A001222(a(n)).

For n > 1, A086971(a(n)) > 1. - Reinhard Zumkeller, Dec 14 2012

Extensions

Name changed by Wesley Ivan Hurt, Apr 25 2020