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A030229 Product of an even number of distinct primes (including 1). 56
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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 (A104141). - Amiram Eldar, May 22 2020

REFERENCES

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995

Ramanujan, Collected Papers, pp. xxiv, 21.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

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

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

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 *)

PROG

(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

(Haskell)

import Data.List (elemIndices)

a030229 n = a030229_list !! (n-1)

a030229_list = map (+ 1) $ elemIndices 1 a008683_list

-- Reinhard Zumkeller, Dec 27 2012

CROSSREFS

Cf. A000379, A005117, A006881, A008683, A028260, A030059, A151797, A245630.

Sequence in context: A339561 A346014 A006881 * A334342 A201650 A201514

Adjacent sequences:  A030226 A030227 A030228 * A030230 A030231 A030232

KEYWORD

nonn,easy,nice

AUTHOR

David W. Wilson

STATUS

approved

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Last modified August 3 07:58 EDT 2021. Contains 346435 sequences. (Running on oeis4.)