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

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

Offset: 1

Author

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

Keywords

• nonn

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

Examples

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

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), 388-392.
• Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

Crossrefs

Subsequence of A085736; A256617 is a subsequence.

Row 2 of A125666.

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.

Cf. A006881, A046386, A046387, A067885 (product of exactly 2, 4, 5, 6 distinct primes respectively).

Cf. A000040, A112801.

Programs

Haskell

a007774 n = a007774_list !! (n-1)
a007774_list = filter ((== 2) . a001221) [1..]
-- Reinhard Zumkeller, Aug 02 2012

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 *)

PARI

is(n)=omega(n)==2 \\ Charles R Greathouse IV, Apr 01 2013

Python

from sympy import primefactors
A007774_list = [n for n in range(1,10**5) if len(primefactors(n)) == 2] # Chai Wah Wu, Aug 23 2021

Extensions

Expanded definition. - N. J. A. Sloane, Aug 22 2021