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A001969 Evil numbers: numbers with an even number of 1's in their binary expansion.
(Formerly M2395 N0952)
188
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; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 français: les nombres païens.

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

REFERENCES

Allouche, Jean-Paul and Cohen, Henri, Dirichlet series and curious infinite products, Bull. London Math. Soc. 17 (1985), 531-538.

Fouvry, E.; Mauduit, C. Sommes des chiffres et nombres presque premiers. (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029)

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.

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.

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).

LINKS

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.-P. Allouche, B. Cloitre, V. Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214, 2014

J. N. Cooper, D. Eichhorn and K. O'Bryant, Reciprocals of binary power series

J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.

Jeffrey O. Shallit, On infinite products associated with sums of digits, J. Number Theory 21 (1985), 128-134.

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

Eric Weisstein's World of Mathematics, Evil Number

Index entries for sequences related to binary expansion of n

Index entries for "core" sequences

FORMULA

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). - Franklin T. Adams-Watters, Aug 28 2006

MAPLE

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.

MATHEMATICA

Select[Range[0, 300], EvenQ[DigitCount[ #, 2][[1]]] &]

PROG

(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

CROSSREFS

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.

For primes see A027699, also A130593.

Cf. A059010.

Sequence in context: A165740 A241571 A080307 * A075311 A032786 A080309

Adjacent sequences:  A001966 A001967 A001968 * A001970 A001971 A001972

KEYWORD

easy,core,nonn,nice,base

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Robin Trew (trew(AT)hcs.harvard.edu).

STATUS

approved

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Last modified September 30 19:51 EDT 2014. Contains 247475 sequences.