A105441 Numbers with at least two odd prime factors (not necessarily distinct).

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141

Offset: 1

Author

Reinhard Zumkeller, Apr 09 2005

Keywords

• nonn
• easy

Comments

Also polite numbers (A138591) that can be expressed as the sum of two or more consecutive integers in more than one ways. For example 9=4+5 and 9=2+3+4. Also 15=7+8, 15=4+5+6 and 15=1+2+3+4+5. - Jayanta Basu, Apr 30 2013

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A093641; A093642 is a subsequence.

Cf. A001227, A087436, A138591.

Programs

Haskell

a105441 n = a105441_list !! (n-1)
a105441_list = filter ((> 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

Mathematica

opf3Q[n_]:=Count[Flatten[Table[First[#],{Last[#]}]&/@FactorInteger[n]], ?OddQ]>1 (* _Harvey P. Dale, Jun 13 2011 *)

PARI

upTo(lim)=my(v=List(),p=7,m);forprime(q=8,lim,forstep(n=p+2,q-2,2,m=n;while(m<=lim,listput(v,m);m<<=1));p=q);forstep(n=p+2,lim,2,listput(v,n));vecsort(Vec(v)) \\ Charles R Greathouse IV, Aug 08 2011

PARI

is(n)=n>>=valuation(n,2); !isprime(n) && n>1 \\ Charles R Greathouse IV, Apr 30 2013

Python

from sympy import primepi
def A105441(n):
    def f(x): return int(n+1+sum(primepi(x>>i) for i in range(x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 02 2025

Formula

A087436(a(n)) > 1.

A001227(a(n)) > 2. [Reinhard Zumkeller, May 01 2012]