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 A030229 Numbers that are the product of an even number of distinct primes. 69
 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 (cf. 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 S. Ramanujan, Collected Papers, pp. xxiv, 21. H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6. 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 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, A104141, A151797, A245630. Sequence in context: A350486 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 February 27 21:03 EST 2024. Contains 370378 sequences. (Running on oeis4.)