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A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

%I A007774 #69 Apr 14 2024 03:49:31
%S A007774 6,10,12,14,15,18,20,21,22,24,26,28,33,34,35,36,38,39,40,44,45,46,48,
%T A007774 50,51,52,54,55,56,57,58,62,63,65,68,69,72,74,75,76,77,80,82,85,86,87,
%U A007774 88,91,92,93,94,95,96,98,99,100,104,106,108,111,112,115,116,117,118
%N A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.
%C A007774 Every group of order p^a * q^b is solvable (Burnside, 1904). - _Franz Vrabec_, Sep 14 2008
%C A007774 Characteristic function for a(n): floor(omega(n)/2) * floor(2/omega(n)) where omega(n) is the number of distinct prime factors of n. - _Wesley Ivan Hurt_, Jan 10 2013
%H A007774 T. D. Noe, <a href="/A007774/b007774.txt">Table of n, a(n) for n = 1..1000</a>
%H A007774 W. Burnside, <a href="https://doi.org/10.1112/plms/s2-1.1.388">On groups of order p^alpha q^beta</a>, Proc. London Math. Soc. (2) 1 (1904), 388-392.
%H A007774 Hans Montanus and Ron Westdijk, <a href="https://greenbluemath.nl/wp-content/uploads/2024/03/Cellular-Automation-and-Binomials.pdf">Cellular Automation and Binomials</a>, Green Blue Mathematics (2022), p. 90.
%e A007774 20 is a term because 20 = 2^2*5 with two distinct prime divisors 2, 5.
%p A007774 with(numtheory,factorset):f := proc(n) if nops(factorset(n))=2 then RETURN(n) fi; end;
%t A007774 Select[Range[0,6! ],Length[FactorInteger[ # ]]==2&] (* _Vladimir Joseph Stephan Orlovsky_, Apr 22 2010 *)
%t A007774 Select[Range[120],PrimeNu[#]==2&] (* _Harvey P. Dale_, Jun 03 2020 *)
%o A007774 (Haskell)
%o A007774 a007774 n = a007774_list !! (n-1)
%o A007774 a007774_list = filter ((== 2) . a001221) [1..]
%o A007774 -- _Reinhard Zumkeller_, Aug 02 2012
%o A007774 (PARI) is(n)=omega(n)==2 \\ _Charles R Greathouse IV_, Apr 01 2013
%o A007774 (Python)
%o A007774 from sympy import primefactors
%o A007774 A007774_list = [n for n in range(1,10**5) if len(primefactors(n)) == 2] # _Chai Wah Wu_, Aug 23 2021
%Y A007774 Subsequence of A085736; A256617 is a subsequence.
%Y A007774 Row 2 of A125666.
%Y A007774 Cf. A001358 (products of two primes), A014612 (products of three primes), A014613 (products of four primes), A014614 (products of five primes), where the primes are not necessarily distinct.
%Y A007774 Cf. A006881, A046386, A046387, A067885 (product of exactly 2, 4, 5, 6 distinct primes respectively).
%Y A007774 Cf. A000040, A112801.
%K A007774 nonn
%O A007774 1,1
%A A007774 Luke Pebody (ltp1000(AT)hermes.cam.ac.uk)
%E A007774 Expanded definition. - _N. J. A. Sloane_, Aug 22 2021