

A007774


Numbers that are divisible by exactly 2 different primes.


51



6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118
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OFFSET

1,1


COMMENTS

Every group of order p^a * q^b is solvable (Burnside, 1904).  Franz Vrabec, Sep 14 2008
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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
W. Burnside, On groups of order p^alpha q^beta, Proc. London Math. Soc. (2) 1 (1904), 388392.


FORMULA

omega(a(n)) = A001221(a(n)) = 2.  Jonathan Vos Post, Sep 20 2005


EXAMPLE

20 is a term because 20 = 2^2*5 with two distinct prime divisors 2, 5.


MAPLE

with(numtheory, factorset):f := proc(n) if nops(factorset(n))=2 then RETURN(n) fi; end;


MATHEMATICA

Select[Range[0, 6! ], Length[FactorInteger[ # ]]==2&] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)
Select[Range[120], PrimeNu[#]==2&] (* Harvey P. Dale, Jun 03 2020 *)


PROG

(Haskell)
a007774 n = a007774_list !! (n1)
a007774_list = filter ((== 2) . a001221) [1..]
 Reinhard Zumkeller, Aug 02 2012
(PARI) is(n)=omega(n)==2 \\ Charles R Greathouse IV, Apr 01 2013


CROSSREFS

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.
See also A074969, A051270, A033993, A033992.
Cf. A001358, A014612, A014613, A014614, A074969, A051270, A033993, A033992, A000040.
Cf. A112801.
Cf. A006881, A046386, A046387, A067885 (product of exactly 2, 4, 5, 6 distinct primes respectively).
Subsequence of A085736.  Franz Vrabec, Sep 14 2008
Cf. A256617 (subsequence).
Sequence in context: A106543 A324455 A327476 * A030231 A267114 A275665
Adjacent sequences: A007771 A007772 A007773 * A007775 A007776 A007777


KEYWORD

nonn


AUTHOR

Luke Pebody (ltp1000(AT)hermes.cam.ac.uk)


STATUS

approved



