A277808 -id:A277808 - cp's OEIS frontend

A129771 Evil odd numbers.

3, 5, 9, 15, 17, 23, 27, 29, 33, 39, 43, 45, 51, 53, 57, 63, 65, 71, 75, 77, 83, 85, 89, 95, 99, 101, 105, 111, 113, 119, 123, 125, 129, 135, 139, 141, 147, 149, 153, 159, 163, 165, 169, 175, 177, 183, 187, 189, 195, 197, 201, 207, 209, 215, 219, 221, 225, 231, 235

Offset: 1

Tanya Khovanova, May 16 2007

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

Francisco J. Muñoz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Francisco J. Muñoz and Juan Carlos Nuño, Rule-based Generation of de Bruijn Sequences: Memory and Learning, arXiv:2507.09764 [cs.FL], 2025. See p. 9.

Intersection of A001969 and A005408.
Supersequence of A093688.
Cf. A092246 (odd odious numbers).
Column 2 of A277880, positions of 1's in A277808 (2's in A277822).
Cf. A000069, A128309, A277823, A277902.

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

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

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

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

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.
A132680(a(n)) = A132680((a(n)-1)/2) + 2. - Reinhard Zumkeller, Aug 26 2007
a(n) = 4n + O(1). - Charles R Greathouse IV, Mar 21 2013
a(n) = A001969(1+A000069(n)) = A277902(A277823(n)). - Antti Karttunen, Nov 05 2016

More terms from Stefan Steinerberger, May 17 2007

A277822 a(n) = index of the column where n is located in array A277880.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 2, 3, 1, 4, 1, 1, 2, 5, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 4, 1, 1, 2, 5, 1, 1, 2, 1, 2, 3, 1, 6, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 2, 3, 1, 4, 1, 1, 2, 5, 1, 1, 2, 1, 2, 3, 1, 6, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 4, 1, 1, 2, 7, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 2, 3, 1, 4, 1, 1, 2, 5

Offset: 0

Antti Karttunen, Nov 03 2016

Ordinal transform of A277813.

a(n) = 1 + the number of iterations of map k -> A003188(A006068(k)/2) that are required (when starting from k = n) until k is an odious number.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

One more than A277808.

Cf. A000069, A001969, A001511, A003188, A006068, A010059, A010060, A245710, A277813, A277818, A277880.

a(0) = 0, for n >= 1, a(n) = 1 + (A010059(n)*A001511(n)).
a(0) = 0, for n >= 1, if A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = 1, otherwise a(n) = 1 + a(A003188(A006068(n)/2)).
Other identities. For all n >= 1:
a(n) = 1 + a(floor(n/2)) when A010060(n) = 0.
a(n) = 1+A277808(n).

A277812 a(n) = the first odious number encountered when starting from k = n and iterating the map k -> A003188(A006068(k)/2).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 8, 4, 2, 11, 1, 13, 14, 7, 16, 8, 4, 19, 2, 21, 22, 11, 1, 25, 26, 13, 28, 14, 7, 31, 32, 16, 8, 35, 4, 37, 38, 19, 2, 41, 42, 21, 44, 22, 11, 47, 1, 49, 50, 25, 52, 26, 13, 55, 56, 28, 14, 59, 7, 61, 62, 31, 64, 32, 16, 67, 8, 69, 70, 35, 4, 73, 74, 37, 76, 38, 19, 79, 2, 81, 82, 41, 84, 42, 21, 87, 88, 44, 22, 91, 11, 93, 94, 47, 1, 97, 98, 49, 100

Offset: 1

Antti Karttunen, Nov 03 2016

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000069, A001969, A000265, A003188, A006068, A010060, A268669, A277813.
Cf. A277808 (gives the number of such iterations needed to reach a(n) from n).
Cf. A003945 (the positions of 1's in this sequence).

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = n, otherwise a(n) = a(A003188(A006068(n)/2)).
Other identities:
a(n) = A000069(A277813(n)).
If A010060(n) = 0 [when n is one of the evil numbers, A001969], then a(n)= a(A000265(n)) [the trailing zeros in binary expansion of n do not affect the result].
For all n >= 1, a(A000069(n)) = A000069(n). [By definition].
For all n > 1, a(A001969(n)) < A001969(n).

A324379 a(n) = A007814(A005187(n)).

Original entry on oeis.org

0, 0, 2, 0, 3, 1, 0, 0, 4, 1, 0, 1, 0, 0, 1, 0, 5, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 0, 0, 6, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 1, 0, 1, 3, 0, 0, 1, 4, 0, 2, 0, 0, 3, 0, 7, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 1, 0, 1, 3, 0, 0, 1, 4, 0, 2, 0, 0, 3, 1, 0, 0, 1, 0, 1, 3, 0, 0, 1

Offset: 1

Antti Karttunen, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A005187, A007814, A277808, A324377.

A324379(n) = { my(s=n); while(n>>=1, s+=n); valuation(s,2); };

a(n) = A007814(A005187(n)).

cp's OEIS Frontend

A129771 Evil odd numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A277822 a(n) = index of the column where n is located in array A277880.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A277812 a(n) = the first odious number encountered when starting from k = n and iterating the map k -> A003188(A006068(k)/2).

Views

Author

Keywords

Links

Crossrefs

Formula

A324379 a(n) = A007814(A005187(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula