|
%I #42 Dec 31 2023 10:21:54
%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 (Sage)
%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
|
