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A001969
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Evil numbers: numbers with an even number of 1's in their binary expansion.
(Formerly M2395 N0952)
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121
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0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 129
(list;
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listen;
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internal format)
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OFFSET
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1,2
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COMMENTS
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This sequence and A000069 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.
En francais: les nombres paiens.
a(n)-A001285(n) = 2n-1 has been verified for n=0,1,2,...,400. - John W. Layman
First differences give A036585. Observed by Franklin T. Adams-Watters, proved by Max Alekseyev, Aug 30 2006. This is equivalent to a(n) = 2*n + A010060(n). Proof: If the number of bits in n is odd then the last bit of a(n) is 1 and if the number of bits in n is even then the last bit of a(n) is 0. Hence the sequence of last bits is A010060. Therefore a(n) = 2*n + A010060(n).
Indices of zeros in the Thue-Morse sequence A010060. - Tanya Khovanova, Feb 13 2009
A005590(a(n)) <= 0. [Reinhard Zumkeller, Apr 11 2012]
A106400(a(n)) = 1. [Reinhard Zumkeller, Apr 29 2012]
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REFERENCES
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Allouche, Jean-Paul and Cohen, Henri, Dirichlet series and curious infinite products, Bull. London Math. Soc. 17 (1985), 531-538.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.
M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
D. J. Newman, A Problem Seminar, Springer; see Problem #89.
Shallit, J. O., On infinite products associated with sums of digits, J. Number Theory 21 (1985), 128-134.
V. S. Shevelev, On some identities connected with the partition of the positive integers with respect to the Morse sequence, Izv. Vuzov of the North-Caucasus region, Nature sciences 4 (1997), 21-23.
VLADIMIR SHEVELEV AND PETER J. C. MOSES, Tangent power sums and their applications, Arxiv preprint arXiv:1207.0404, 2012. - From N. J. A. Sloane, Dec 17 2012
V. Shevelev and P. J. C. Moses, A family of digit functions with large periods, arXiv preprint arXiv:1209.5705, 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J. N. Cooper, D. Eichhorn and K. O'Bryant, Reciprocals of binary power series
Eric Weisstein's World of Mathematics, Evil Number
Index entries for sequences related to binary expansion of n
Index entries for "core" sequences
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FORMULA
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Note that 2n+1 is in the sequence iff 2n is not and so this sequence has asymptotic density 1/2. - Franklin T. Adams-Watters, Aug 23 2006
a(n) = (1/2) * (4n + 1 - (-1)^A000120(n)). - Ralf Stephan, Sep 14 2003
G.f.: sum[k>=0, t(3+2t+3t^2)/(1-t^2)^2 * prod(l=0, k-1, 1-x^(2^l)), t=x^2^k]. - Ralf Stephan, Mar 25 2004
n such that A010060(n)=0. - Benoit Cloitre, Nov 15 2003
a(2*n+1) + a(2*n) = A017101(n) = 8*n+3. a(2*n+1) - a(2*n) gives the Thue-Morse sequence (3, 1 version): 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, .... A001969(n) + A000069(n) = A016813(n) = 4*n+1. - Philippe Deléham, Feb 04 2004
a(1) = 0; for n>1: a(n) = 3*n-3 -a(n/2) if n even, a(n) = a((n+1)/2)+n-1 if n odd.
Let b(n) = 1 if sum of digits of n is even, -1 if it is odd; then Shallit (1985) showed that Prod_{n>=0} ((2n+1)/(2n+2))^b(n) = 1/sqrt(2).
a(n) = 2n + A010060(n). - Frank Adams-Watters, Aug 28 2006
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MAPLE
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s := proc(n) local i, j, ans; ans := [ ]; j := 0; for i from 0 while j<n do if add(k, k=convert(i, base, 2)) mod 2=0 then ans := [ op(ans), i ]; j := j+1; fi; od; RETURN(ans); end; t1 := s(100); A001969 := n->t1[n]; # s(k) gives first k terms.
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MATHEMATICA
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Select[Range[0, 300], EvenQ[DigitCount[ #, 2][[1]]] &]
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PROG
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(PARI) a(n)=2*n+subst(Pol(binary(n)), x, 1)%2
(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)+n, -a((n-1)/2)+3*n))
(PARI) a(n)=2*n--+hammingweight(n)%2 \\ Charles R Greathouse IV, Mar 22 2013
(MAGMA) [ n : n in [0..129] | IsEven(&+Intseq(n, 2)) ]; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(Haskell)
a001969 n = a001969_list !! (n-1)
a001969_list = [x | x <- [0..], even $ a000120 x]
-- Reinhard Zumkeller, Feb 01 2012
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CROSSREFS
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Complement of A000069 (the odious numbers). Cf. A133009.
a(n)=2*n+A010060(n)=A000069(n)-(-1)^A010060(n). Cf. A018900.
The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.
Cf. A036585 (differences), A010060, A006364.
Sequence in context: A187837 A165740 A080307 * A075311 A032786 A080309
Adjacent sequences: A001966 A001967 A001968 * A001970 A001971 A001972
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KEYWORD
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easy,core,nonn,nice,base
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Robin Trew (trew(AT)hcs.harvard.edu).
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STATUS
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approved
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