A102467 - OEIS

%I #43 Sep 22 2025 16:00:46

%S 1,12,18,20,24,28,30,36,40,42,44,45,48,50,52,54,56,60,63,66,68,70,72,

%T 75,76,78,80,84,88,90,92,96,98,99,100,102,104,105,108,110,112,114,116,

%U 117,120,124,126,130,132,135,136,138,140,144,147,148,150,152,153,154,156

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

%C These are the numbers which are neither prime powers (>1) nor semiprimes. - _M. F. Hasler_, Jan 31 2008

%C 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

%H T. D. Noe, <a href="/A102467/b102467.txt">Table of n, a(n) for n = 1..1000</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 40.4.1.3 "Testing for irreducibility without GCD computations", pp. 839-840.

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

%F For n > 1, A086971(a(n)) > 1. - _Reinhard Zumkeller_, Dec 14 2012

%e 10 is not in the sequence since d(10) = 4 is equal to Omega(10) + omega(10) = 2 + 2 = 4.

%e 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

%p with(numtheory):

%p q:= n-> is(tau(n)<>bigomega(n)+nops(factorset(n))):

%p select(q, [$1..200])[];  # _Alois P. Heinz_, Jul 14 2023

%t Select[Range[200], DivisorSigma[0, #] != PrimeOmega[#] + PrimeNu[#]&] (* _Jean-François Alcover_, Jun 22 2018 *)

%o (SageMath)

%o def is_A102467(n) :

%o     return sloane.A001221(n) != 1 and sloane.A001222(n) != 2

%o def A102467_list(n) :

%o     return [k for k in (1..n) if is_A102467(k)]

%o A102467_list(156)  # _Peter Luschny_, Feb 07 2012

%o (Haskell)

%o a102467 n = a102467_list !! (n-1)

%o a102467_list = [x | x <- [1..], a000005 x /= a001221 x + a001222 x]

%o -- _Reinhard Zumkeller_, Dec 14 2012

%o (PARI) is(n)=my(f=factor(n)[,2]); #f!=1 && f!=[1,1]~ \\ _Charles R Greathouse IV_, Oct 19 2015

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

%Y Cf. A102466, A135767.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jan 09 2005

%E Name changed by _Wesley Ivan Hurt_, Apr 25 2020