A277813 -id:A277813 - cp's OEIS frontend

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

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).

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)).

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).

A277881 Inverse permutation to A277880.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 10, 15, 9, 8, 21, 7, 28, 36, 14, 45, 20, 13, 55, 12, 66, 78, 27, 11, 91, 105, 35, 120, 44, 19, 136, 153, 54, 26, 171, 18, 190, 210, 65, 17, 231, 253, 77, 276, 90, 34, 300, 16, 325, 351, 104, 378, 119, 43, 406, 435, 135, 53, 465, 25, 496, 528, 152, 561, 170, 64, 595, 33, 630, 666, 189, 24, 703, 741, 209, 780, 230, 76, 820, 23

Offset: 1

Antti Karttunen, Nov 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A277880.
Cf. A277813, A277822.
Cf. A028401 (terms at powers of 2).

(define (A277881 n) (let ((row (A277813 n)) (col (A277822 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A277822(n), and r = A277813(n).

A277883 Inverse permutation to A277882.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 11, 8, 9, 16, 10, 22, 29, 12, 37, 17, 13, 46, 14, 56, 67, 23, 15, 79, 92, 30, 106, 38, 18, 121, 137, 47, 24, 154, 19, 172, 191, 57, 20, 211, 232, 68, 254, 80, 31, 277, 21, 301, 326, 93, 352, 107, 39, 379, 407, 122, 48, 436, 25, 466, 497, 138, 529, 155, 58, 562, 32, 596, 631, 173, 26, 667, 704, 192, 742, 212, 69, 781, 27

Offset: 1

Antti Karttunen, Nov 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A277882.

Cf. A277813, A277822.

(define (A277883 n) (let ((col (A277813 n)) (row (A277822 n))) (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))

a(n) = (1/2) * ((c+r)^2 - r - 3*c + 2), where c = A277813(n), and r = A277822(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

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

A277881 Inverse permutation to A277880.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A277883 Inverse permutation to A277882.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula