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A030229
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Product of an even number of distinct primes (including 1).
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24
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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;
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995
Ramanujan, Collected Papers, pp. xxiv, 21.
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LINKS
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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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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Select[Range[214], MoebiusMu[#] == 1 &] (* Jean-François Alcover, Oct 04 2011 *)
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PROG
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(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
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CROSSREFS
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Cf. A006881, A151797, A030059, A005117, A008683.
Sequence in context: A265693 A211484 A006881 * A201650 A201514 A201464
Adjacent sequences: A030226 A030227 A030228 * A030230 A030231 A030232
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KEYWORD
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nonn,easy,nice
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AUTHOR
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David W. Wilson
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STATUS
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approved
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