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A102466 Numbers such that the number of divisors is the sum of numbers of prime factors with and without repetitions.

%I A102466 #21 Jul 14 2023 15:30:35
%S A102466 2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,31,32,
%T A102466 33,34,35,37,38,39,41,43,46,47,49,51,53,55,57,58,59,61,62,64,65,67,69,
%U A102466 71,73,74,77,79,81,82,83,85,86,87,89,91,93,94,95,97,101,103,106,107,109
%N A102466 Numbers such that the number of divisors is the sum of numbers of prime factors with and without repetitions.
%C A102466 A000005(a(n)) = A001221(a(n)) + A001222(a(n)); prime powers are a subsequence (A000961); complement of A102467; not the same as A085156.
%C A102466 Equals { n | omega(n)=1 or Omega(n)=2 }, that is, these are exactly the prime powers (>1) and semiprimes. - _M. F. Hasler_, Jan 14 2008
%C A102466 For n > 1: A086971(a(n)) <= 1. - _Reinhard Zumkeller_, Dec 14 2012
%H A102466 T. D. Noe, <a href="/A102466/b102466.txt">Table of n, a(n) for n = 1..1000</a>
%p A102466 with(numtheory):
%p A102466 q:= n-> is(tau(n)=bigomega(n)+nops(factorset(n))):
%p A102466 select(q, [$1..200])[];  # _Alois P. Heinz_, Jul 14 2023
%t A102466 Select[Range[110],DivisorSigma[0,#]==PrimeOmega[#]+PrimeNu[#]&] (* _Harvey P. Dale_, Mar 09 2016 *)
%o A102466 (Sage)
%o A102466 def is_A102466(n) :
%o A102466     return bool(sloane.A001221(n) == 1 or sloane.A001222(n) == 2)
%o A102466 def A102466_list(n) :
%o A102466     return [k for k in (1..n) if is_A102466(k)]
%o A102466 A102466_list(109)  # Peter Luschny, Feb 08 2012
%o A102466 (Haskell)
%o A102466 a102466 n = a102466_list !! (n-1)
%o A102466 a102466_list = [x | x <- [1..], a000005 x == a001221 x + a001222 x]
%o A102466 -- _Reinhard Zumkeller_, Dec 14 2012
%o A102466 (PARI) is(n)=my(f=factor(n)[,2]); #f==1 || f==[1,1]~ \\ _Charles R Greathouse IV_, Oct 19 2015
%Y A102466 Cf. A000005, A001221, A001222, A000961, A102467, A085156,  A086971, A135767.
%K A102466 nonn
%O A102466 1,1
%A A102466 _Reinhard Zumkeller_, Jan 09 2005