A030229 Numbers that are the product of an even number of distinct primes.

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 210, 213, 214

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy
• nice

Comments

These are the positive integers k with moebius(k) = 1 (cf. A008683). - N. J. A. Sloane, May 18 2021

From Enrique Pérez Herrero, Jul 06 2012: (Start)

This sequence and A030059 form a partition of the squarefree numbers set: A005117.

Also solutions to equation mu(n)=1.

Sum_{n>=1} 1/a(n)^s = (Zeta(s)^2 + Zeta(2*s))/(2*Zeta(s)*Zeta(2*s)).

(End)

A008683(a(n)) = 1; a(A220969(n)) mod 2 = 0; a(A220968(n)) mod 2 = 1. - Reinhard Zumkeller, Dec 27 2012

Characteristic function for values of a(n) = (mu(n)+1)! - 1, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Oct 11 2013

Conjecture: For the matrix M(i,j) = 1 if j|i and 0 otherwise, Inverse(M)(a,1) = -1, for any a in this sequence. - Benedict W. J. Irwin, Jul 26 2016

Solutions to the equation Sum_{d|n} mu(d)*d = Sum_{d|n} mu(n/d)*d. - Torlach Rush, Jan 13 2018

Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = n, where sigma(n) is the sum of divisors function (A000203). - Robert D. Rosales, May 20 2024

From Peter Munn, Oct 04 2019: (Start)

Numbers n such that omega(n) = bigomega(n) = 2*k for some integer k.

The squarefree numbers in A000379.

The squarefree numbers in A028260.

This sequence is closed with respect to the commutative binary operation A059897(.,.), thus it forms a subgroup of the positive integers under A059897(.,.). A006094 lists a minimal set of generators for this subgroup. The lexicographically earliest ordered minimal set of generators is A100484 with its initial 4 removed.

(End)

The asymptotic density of this sequence is 3/Pi^2 (cf. A104141). - Amiram Eldar, May 22 2020

Examples

			(empty product), 2*3, 2*5, 2*7, 3*5, 3*7, 2*11, 2*13, 3*11, 2*17, 5*7, 2*19, 3*13, 2*23,...
		

References

• B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995
• S. Ramanujan, Collected Papers, pp. xxiv, 21.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Debmalya Basak, Nicolas Robles, and Alexandru Zaharescu, Exponential sums over Möbius convolutions with applications to partitions, arXiv:2312.17435 [math.NT], 2023. Mentions this sequence.
• S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
• Eric Weisstein's World of Mathematics, Prime Factor
• Eric Weisstein's World of Mathematics, Moebius Function
• Eric Weisstein's World of Mathematics, Prime Sums
• H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6.

Crossrefs

A006881, A046386, A067885, A123322, A281222 are subsequences.

Cf. A000379, A005117, A008683, A028260, A030059, A104141, A151797, A245630.

Programs

Haskell

import Data.List (elemIndices)
a030229 n = a030229_list !! (n-1)
a030229_list = map (+ 1) $ elemIndices 1 a008683_list
-- Reinhard Zumkeller, Dec 27 2012

Maple

a := n -> `if`(numtheory[mobius](n)=1,n,NULL); seq(a(i),i=1..214); # Peter Luschny, May 04 2009
with(numtheory); t := [ ]: f := [ ]: for n from 1 to 250 do if mobius(n) = 1 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # Wesley Ivan Hurt, Oct 11 2013
# alternative
A030229 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if numtheory[mobius](a) = 1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A030229(n),n=1..40) ; # R. J. Mathar, Sep 22 2020

Mathematica

Select[Range[214], MoebiusMu[#] == 1 &] (* Jean-François Alcover, Oct 04 2011 *)

PARI

isA030229(n)= #(n=factor(n)[,2]) % 2 == 0 && (!n || vecmax(n)==1 )

PARI

is(n)=moebius(n)==1 \\ Charles R Greathouse IV, Jan 31 2017
for(n=1,500, isA030229(n)&print1(n",")) \\ M. F. Hasler

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A030229(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,x.bit_length(),2)))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024

Formula

a(n) < n*Pi^2/3 infinitely often; a(n) > n*Pi^2/3 infinitely often. - Charles R Greathouse IV, Oct 04 2011; corrected Sep 07 2017

{a(n)} = {m : m = A059897(A030059(k), p), k >= 1} for prime p, where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Oct 04 2019