A120944 Composite squarefree numbers.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Author

Zak Seidov, Aug 19 2006

Keywords

• nonn

Comments

Intersection of A002808 and A005117: n > 1 such that A008966(n) * (1-A010051(n)) = 1. - Reinhard Zumkeller, Dec 19 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000469 (Nonprime squarefree numbers).

Cf. A001221, A001222, A002808, A005117, A008966, A010051.

Set of powers: A182853.

Programs

Haskell

a120944 n = a120944_list !! (n-1)
a120944_list = filter ((== 1) . a008966) a002808_list
-- Reinhard Zumkeller, Dec 19 2011

Magma

[n: n in [6..161] | IsSquarefree(n) and not IsPrime(n)]; // Bruno Berselli, Mar 03 2011

Maple

select(not(isprime) and numtheory:-issqrfree, [$2..1000]); # Robert Israel, Jul 07 2015

Mathematica

lst={};Do[If[SquareFreeQ[n],If[ !PrimeQ[n],AppendTo[lst,n]]],{n,2,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009; updated by Jean-François Alcover, Jun 19 2013 *)
Select[Range[200], PrimeNu[#] > 1 && SquareFreeQ[#] &] (* Carlos Eduardo Olivieri, Jul 07 2015 *)

PARI

is(n)=issquarefree(n)&&!isprime(n)&&n>1 \\ Charles R Greathouse IV, Apr 11 2012

Python

from sympy import factorint
def ok(n): f = factorint(n); return len(f) > 1 and all(f[p] < 2 for p in f)
print(list(filter(ok, range(1, 162)))) # Michael S. Branicky, Jun 10 2021

Python

from math import isqrt
from sympy import primepi, mobius
def A120944(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 02 2024

Formula

From Enrique Pérez Herrero, Apr 01 2012: (Start)

Solutions to floor(omega(x)/bigomega(x))*(1-floor(1/bigomega(x))) = 1, where bigomega is A001222 and omega is A001221.

Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(2s) - 1 - PrimeZeta(s). (End)

a(n) = kn + O(n/log n) where k = Pi^2/6. - Charles R Greathouse IV, Aug 02 2024