OFFSET

1,1

COMMENTS

A heuristic argument suggests that, as n tends to infinity, a(n)/n converges to 4. - Stefan Steinerberger, May 17 2007

These numbers may be called primitive evil numbers because every evil number is a power of 2 multiplied by one of these numbers. Note that the difference between consecutive terms is either 2, 4, or 6. - T. D. Noe, Jun 06 2007

If m is in the sequence, then so is 2m-1 because in binary, m is x1 and 2m-1 is x01. Presumably the numbers that generate the whole sequence by application of n -> 2n-1 are the evil numbers times 4 plus 3. - Ralf Stephan, May 25 2013

LINKS

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

FORMULA

a(n) = 2*A000069(n) + 1. a(n) is 1 plus twice odious numbers. a(n) = A128309(n) + 1. a(n) is 1 plus odious even numbers.

a(n) = 4n + O(1). - Charles R Greathouse IV, Mar 21 2013

MATHEMATICA

Select[Range[300], OddQ[ # ] && EvenQ[DigitCount[ #, 2, 1]] &] (* Stefan Steinerberger, May 17 2007 *)

Select[Range[300], EvenQ[Plus @@ IntegerDigits[ #, 2]] && OddQ[ # ] &]

PROG

(PARI) is(n)=n%2 && hammingweight(n)%2==0 \\ Charles R Greathouse IV, Mar 21 2013

(PARI) a(n)=4*n-if(hammingweight(n-1)%2, 3, 1) \\ Charles R Greathouse IV, Mar 21 2013

(Python)

def A129771(n): return (((m:=n-1)<<1)+(m.bit_count()&1^1)<<1)+1 # Chai Wah Wu, Mar 09 2023

CROSSREFS

This sequence is the intersection of A001969 (Evil numbers: even number of 1's in binary expansion.) and A005408 (The odd numbers: a(n) = 2n+1.) A093688 (Numbers n such that all divisors of n, excluding the divisor 1, have an even number of 1's in their binary expansions) is a subsequence.

Cf. A092246 (odd odious numbers).

KEYWORD

nonn,easy

AUTHOR

Tanya Khovanova, May 16 2007

EXTENSIONS

More terms from Stefan Steinerberger, May 17 2007

STATUS

approved