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

1,1

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), 388-392.

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

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

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.

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)

Extensions

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

Status

approved