A277902 -id:A277902 - 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

A277880 Dispersion of evil numbers: Square array A(r,c) with A(r,1) = A000069(r); and for c > 1, A(r,c) = A001969(1+(A(r,c-1))), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 12, 10, 9, 7, 24, 20, 18, 15, 8, 48, 40, 36, 30, 17, 11, 96, 80, 72, 60, 34, 23, 13, 192, 160, 144, 120, 68, 46, 27, 14, 384, 320, 288, 240, 136, 92, 54, 29, 16, 768, 640, 576, 480, 272, 184, 108, 58, 33, 19, 1536, 1280, 1152, 960, 544, 368, 216, 116, 66, 39, 21, 3072, 2560, 2304, 1920, 1088, 736, 432, 232, 132, 78, 43, 22

Offset: 1

Antti Karttunen, Nov 03 2016

			The top left 12 x 12 corner of the array:
   1,  3,  6,  12,  24,  48,   96,  192,  384,   768,  1536,  3072
   2,  5, 10,  20,  40,  80,  160,  320,  640,  1280,  2560,  5120
   4,  9, 18,  36,  72, 144,  288,  576, 1152,  2304,  4608,  9216
   7, 15, 30,  60, 120, 240,  480,  960, 1920,  3840,  7680, 15360
   8, 17, 34,  68, 136, 272,  544, 1088, 2176,  4352,  8704, 17408
  11, 23, 46,  92, 184, 368,  736, 1472, 2944,  5888, 11776, 23552
  13, 27, 54, 108, 216, 432,  864, 1728, 3456,  6912, 13824, 27648
  14, 29, 58, 116, 232, 464,  928, 1856, 3712,  7424, 14848, 29696
  16, 33, 66, 132, 264, 528, 1056, 2112, 4224,  8448, 16896, 33792
  19, 39, 78, 156, 312, 624, 1248, 2496, 4992,  9984, 19968, 39936
  21, 43, 86, 172, 344, 688, 1376, 2752, 5504, 11008, 22016, 44032
  22, 45, 90, 180, 360, 720, 1440, 2880, 5760, 11520, 23040, 46080

Inverse permutation: A277881.
Transpose: A277882.
Column 1: A000069, column 2: A129771.
Row 1: A003945.
Cf. A277813 (index of the row where n is located in this array), A277822 (index of the column).
Cf. A001969.
Other related tables or permutations: A277820, A277902, A248513.

(define (A277880 n) (A277880bi (A002260 n) (A004736 n)))
(define (A277880bi row col) (if (= 1 col) (A000069 row) (A001969 (+ 1 (A277880bi row (- col 1))))))
;; Alternatively:
(define (A277880bi row col) (cond ((= 1 col) (A000069 row)) ((= 2 col) (A129771 row)) (else (* 2 (A277880bi row (- col 1))))))
(define (A129771 n) (+ 1 (* 2 (A000069 n))))

A(r,1) = A000069(r) and for c > 1, A(r,c) = A001969(1+(A(r,c-1))).
Alternatively, if we set also the second column explicitly as:
   A(r,2) = A129771(r) = 1+ 2*A000069(r),
then the rest of entries in each row are obtained just by doubling the preceding term on the same row: A(r,c) = 2*A(r,c-1), for c >= 3.
As a composition of other permutations:
a(n) = A277902(A277820(n)).

A277901 If A010060(n) = 1, a(n) = A065621(A115384(n)), otherwise a(n) = A048724(a(floor(n/2))).

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 13, 9, 10, 14, 15, 11, 8, 12, 25, 23, 27, 26, 30, 31, 28, 18, 17, 21, 22, 29, 19, 24, 20, 16, 49, 43, 57, 50, 45, 55, 52, 46, 34, 61, 62, 33, 59, 36, 54, 56, 51, 41, 42, 63, 47, 58, 39, 44, 37, 53, 40, 38, 60, 35, 32, 48, 97, 83, 125, 98, 75, 103, 100, 86, 119, 109, 110, 89, 107, 92, 114

Offset: 1

Antti Karttunen, Nov 03 2016

a(n) gives the number that is in the same position in array A277820 as where n is located in array A277880.

			The top left corner of array A277880 is:
   1,  3,  6, 12
   2,  5, 10, 20
   4,  9, 18, 36
   7, 15, 30, 60
   8, 17, 34, 68
while the top left corner of A277820 is:
   1,  3,  5, 15
   2,  6, 10, 30
   7,  9, 27, 45
   4, 12, 20, 60
  13, 23, 57, 75
thus a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 7, a(5) = 6, a(6) = 5, a(7) = 4, a(8) = 13, a(9) = 9, a(12) = 15 and a(15) = 12.

Inverse: A277902.
Cf. A000069, A001969, A010060, A048724, A065621, A115384, A129771, A277823, A277825.
Related permutations and arrays: A277820, A277880, A277881.

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = A065621(A115384(n)), otherwise a(n) = A048724(a(floor(n/2))).
As a composition of other permutations:
a(n) = A277820(A277881(n)).
Other identities. For all n >= 1:
A010060(a(n)) = A010060(n). [Preserves the parity of binary weight.]
a(A000069(n)) = A065621(n).
a(A003945(n)) = A001317(n).
a(A129771(n)) = A277823(n).
a(2*A129771(n)) = A277825(n).

cp's OEIS Frontend

A129771 Evil odd numbers.

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A277880 Dispersion of evil numbers: Square array A(r,c) with A(r,1) = A000069(r); and for c > 1, A(r,c) = A001969(1+(A(r,c-1))), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A277901 If A010060(n) = 1, a(n) = A065621(A115384(n)), otherwise a(n) = A048724(a(floor(n/2))).

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