A000469 1 together with products of 2 or more distinct primes.

1, 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

Offset: 1

Author

Dan Bentley (dtb(AT)research.att.com)

Keywords

• nonn
• easy
• nice

Comments

Nonprime squarefree numbers.

Except for 1, composite n such that the squarefree part of n is greater than phi(n). - Benoit Cloitre, Apr 06 2002

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A007913, A000010, A010051, A239508, A239509, A120944 (composite squarefree numbers, same sequence apart from the first term).

Programs

Haskell

a000469 n = a000469_list !! (n-1)
a000469_list = filter ((== 0) . a010051) a005117_list
-- Reinhard Zumkeller, Mar 21 2014

Maple

select(numtheory:-issqrfree and not isprime, [$1..1000]); # Robert Israel, Aug 06 2015

Mathematica

lst={}; Do[If[SquareFreeQ[n], If[ !PrimeQ[n], AppendTo[lst,n]]], {n,200}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009 *)
With[{upto=200},Complement[Select[Range[upto],SquareFreeQ],Prime[ Range[ PrimePi[ upto]]]]] (* Harvey P. Dale, Oct 01 2011 *)
Select[Range[200], !PrimeQ[#] && PrimeOmega[#] == PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 06 2015 *)

PARI

for(n=0,64, if(isprime(n), n+1, if(issquarefree(n),print(n))))

PARI

for(n=1,160,if(core(n)*(1-isprime(n))>eulerphi(n),print1(n,",")))

Python

from math import isqrt
from sympy import primepi, mobius
def A000469(n):
    def f(x): return n+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 02 2024

Formula

n such that A007913(n)>A000010(n). - Benoit Cloitre, Apr 06 2002

N-floor(N/p1) - floor(N/(p2) - ... - floor(N/p(i) + floor(N/(c2) + floor(N/(c3)+ ... + floor(N/c(j)-1 where N is any number; p1,p2 are the primes with p(i) being the first prime > square root of N and c2, c3 are the numbers other than 1 in this sequence with c(j) <= N will yield the number of primes less than or equal to N other than p1, p2, ..., p(i). - Ben Paul Thurston, Aug 15 2007

A005171(a(n))*A008966(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009

Sum(n=1, Infinity, 1/a(n)^s) = Zeta(s)/Zeta(2s) - PrimeZeta(s). - Enrique Pérez Herrero, Mar 31 2012

n such that A001221(n) = A001222(n), n nonprime. - Carlos Eduardo Olivieri, Aug 06 2015

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