A209268 -id:A209268 - cp's OEIS frontend

A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 24, 20, 14, 9, 32, 48, 40, 28, 18, 11, 64, 96, 80, 56, 36, 22, 13, 128, 192, 160, 112, 72, 44, 26, 15, 256, 384, 320, 224, 144, 88, 52, 30, 17, 512, 768, 640, 448, 288, 176, 104, 60, 34, 19, 1024, 1536, 1280, 896, 576, 352, 208, 120

Offset: 0

Henry Bottomley, Apr 12 2000

First column of array is powers of 2, first row is odd numbers, other cells are products of these two, so every positive integer appears exactly once. [Comment edited to match the definition. - L. Edson Jeffery, Jun 05 2015]
An analogous N X N <-> N bijection based, not on the binary, but on the Fibonacci number system, is given by the Wythoff array A035513.
As an array, this sequence (hence also A135764) is the dispersion of the even positive integers. For the definition of dispersion, see the link "Interspersions and Dispersions." The fractal sequence of this dispersion is A003602. - Clark Kimberling, Dec 03 2010

			Northwest corner of array A:
    1     3     5     7     9    11    13    15    17    19
    2     6    10    14    18    22    26    30    34    38
    4    12    20    28    36    44    52    60    68    76
    8    24    40    56    72    88   104   120   136   152
   16    48    80   112   144   176   208   240   272   304
   32    96   160   224   288   352   416   480   544   608
   64   192   320   448   576   704   832   960  1088  1216
  128   384   640   896  1152  1408  1664  1920  2176  2432
  256   768  1280  1792  2304  2816  3328  3840  4352  4864
  512  1536  2560  3584  4608  5632  6656  7680  8704  9728
[Array edited to match the definition. - _L. Edson Jeffery_, Jun 05 2015]
From _Philippe Deléham_, Dec 13 2013: (Start)
a(13-1)=20=2*10, so a(13)=10+A006519(20)=10+4=14.
a(3-1)=3=2*1+1, so a(3)=2^(1+1)=4. (End)
From _Wolfdieter Lang_, Jan 30 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    1
   1:    2    3
   2:    4    6    5
   3:    8   12   10   7
   4:   16   24   20  14   9
   5:   32   48   40  28  18  11
   6:   64   96   80  56  36  22  13
   7:  128  192  160 112  72  44  26  15
   8:  256  384  320 224 144  88  52  30 17
   9:  512  768  640 448 288 176 104  60 34 19
  10: 1024 1536 1280 896 576 352 208 120 68 38 21
  ...
T(3, 2) = 2^1*(2*2+1) = 10. (End)

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Clark Kimberling, Interspersions and Dispersions.
Index entries for sequences that are permutations of the natural numbers

The sequence is a permutation of A000027.
Main diagonal is A014480; inverse permutation is A209268.
Cf. A001511, A002024, A003602, A006519, A025480, A035513, A075300, A135764.

a054582 n k = a054582_tabl !! n !! k
a054582_row n = a054582_tabl !! n
a054582_tabl = iterate
   (\xs@(x:_) -> (2 * x) : zipWith (+) xs (iterate (`div` 2) (2 * x))) [1]
a054582_list = concat a054582_tabl
-- Reinhard Zumkeller, Jan 22 2013

(* Array: *)
Grid[Table[2^m*(2*k + 1), {m, 0, 9}, {k, 0, 9}]] (* L. Edson Jeffery, Jun 05 2015 *)
(* Array antidiagonals flattened: *)
Flatten[Table[2^(m - k)*(2*k + 1), {m, 0, 9}, {k, 0, m}]] (* L. Edson Jeffery, Jun 05 2015 *)

T(m,k)=(2*k+1)<Charles R Greathouse IV, Jun 21 2017

As a sequence, if n is a triangular number, then a(n)=a(n-A002024(n))+2, otherwise a(n)=2*a(n-A002024(n)-1).
a(n) = A075300(n-1)+1.
Recurrence for the sequence: if a(n-1)=2*k is even, then a(n)=k+A006519(2*k); if a(n-1)=2*k+1 is odd, then a(n)=2^(k+1), a(0)=1. - Philippe Deléham, Dec 13 2013
m = A(A001511(m)-1, A003602(m)-1), for each m in A000027. - L. Edson Jeffery, Nov 22 2015
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1), for n >= 0 and k = 0..n. - Wolfdieter Lang, Jan 30 2019

Offset corrected by Reinhard Zumkeller, Jan 22 2013

A246676 Permutation of natural numbers: a(n) = A242378(A007814(n), (1+A000265(n))) - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 11, 24, 13, 26, 15, 10, 17, 20, 19, 34, 21, 44, 23, 48, 25, 32, 27, 124, 29, 80, 31, 12, 33, 74, 35, 54, 37, 62, 39, 76, 41, 38, 43, 174, 45, 134, 47, 120, 49, 50, 51, 64, 53, 98, 55, 342, 57, 104, 59, 624, 61, 242, 63, 16, 65, 56, 67, 244, 69, 224, 71, 90, 73, 68

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

To compute a(n) we shift its binary representation right as many steps k as necessary that the result were an odd number. Then one is added to that odd number, and the prime factorization of the resulting even number is shifted the same k number of steps towards larger primes, whose product is then decremented by one to get the final result.
In the essence, a(n) tells which number in array A246275 is at the same position where n is in the array A135764. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n.
Equally: a(n) tells which number in array A246273 is at the same position where n is in the array A054582, as they are the transposes of above two arrays.

			Consider n=36, "100100" in binary. It has to be shifted two bits right that the result were an odd number 9, "1001" in binary. We see that 9+1 = 10 = 2*5 = p_1 * p_3 [where p_k denotes the k-th prime, A000040(k)], and shifting this two steps towards larger primes results p_3 * p_5 = 5*11 = 55, thus a(36) = 55-1 = 54.

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

Inverse: A246675.
Even bisection halved: A246680.
More recursed versions: A246678, A246684.
Other related permutations: A209268, A246273, A246275, A135764, A054582.
Cf. A000040, A000265, A005408, A007814, A003961, A242378.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A246676(n) = { my(k=0); while(!(n%2), n = n\2; k++); n++; while(k>0, n = A003961(n); k--); n-1; };
for(n=1, 8192, write("b246676.txt", n, " ", A246676(n)));

(define (A246676 n) (+ -1 (A242378bi (A007814 n) (+ 1 (A000265 n))))) ;; Code for A242378bi given in A242378.

a(n) = A242378(A007814(n), (1+A000265(n))) - 1. [Where the bivariate function A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n].
As a composition of related permutations:
a(n) = A246273(A209268(n)).
Other identities:
For all n >= 0, a(A005408(n)) = A005408(n). [Fixes the odd numbers].

A249725 Inverse permutation to A135764.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 7, 10, 11, 8, 16, 9, 22, 12, 29, 15, 37, 17, 46, 13, 56, 23, 67, 14, 79, 30, 92, 18, 106, 38, 121, 21, 137, 47, 154, 24, 172, 57, 191, 19, 211, 68, 232, 31, 254, 80, 277, 20, 301, 93, 326, 39, 352, 107, 379, 25, 407, 122, 436, 48, 466, 138, 497, 28, 529, 155, 562, 58, 596, 173, 631, 32, 667, 192, 704, 69, 742, 212, 781, 26, 821, 233, 862, 81

Offset: 1

Antti Karttunen, Nov 15 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A135764.
Similar or related permutations: A209268, A246276, A246676, A249742, A249811.
Cf. A000079, A000124, A000217, A003602, A005408, A007814.

a(n) = {r = 1; while(!(n%2), n = n >> 1; r++); k = 1 + n \ 2; binomial(r + k - 1, 2) + r} \\ David A. Corneth, Feb 05 2015

(define (A249725 n) (+ 1 (* (/ 1 2) (+ (expt (+ (A003602 n) (A007814 n)) 2) (- (A003602 n)) (A007814 n)))))

a(n) = 1 + (((A003602(n)+A007814(n))^2 + A007814(n) - A003602(n))/2).
As a composition of other permutations:
a(n) = A249742(A249811(n)).
a(n) = A246276(A246676(n)).
Other identities. For all n >= 0 the following holds:
a(A005408(n)) = A000124(n). [Maps odd numbers to central polygonal numbers].
a(A000079(n)) = A000217(n+1). [Maps powers of two to triangular numbers].

A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 11, 24, 13, 20, 15, 10, 17, 26, 19, 34, 21, 32, 23, 48, 25, 38, 27, 54, 29, 44, 31, 12, 33, 50, 35, 64, 37, 56, 39, 76, 41, 62, 43, 84, 45, 68, 47, 120, 49, 74, 51, 94, 53, 80, 55, 90, 57, 86, 59, 114, 61, 92, 63, 16, 65, 98, 67, 124, 69, 104, 71, 118, 73, 110, 75, 144, 77, 116, 79, 142, 81

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

In the essence, a(n) tells which number in square array A249741 (the sieve of Eratosthenes minus 1) is at the same position where n is in array A135764, which is formed from odd numbers whose binary expansions are shifted successively leftwards on the successive rows. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.

Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A054582, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249812.
Cf. A000079, A000265, A001511, A003602, A005408, A006093, A007814, A054582, A083140, A083221, A249741.
Similar or related permutations: A249814 ("deep variant"), A246676, A249815, A114881, A209268, A249725, A249741.
Differs from A246676 for the first time at n=14, where a(14)=20, while
A246676(14)=26.

(define (A249811 n) (+ -1 (A083221bi (A001511 n) (A003602 n))))
;; Code for A083221bi given in A083221

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(n) = A083221(A001511(n),A003602(n)) - 1 = A249741(A001511(n),A003602(n)).
As a composition of related permutations:
a(n) = A114881(A209268(n)).
a(n) = A249741(A249725(n)).
a(n) = A249815(A246676(n)).
Other identities. For all n >= 1 the following holds:
a(A000079(n-1)) = A006093(n).

A246274 Inverse of A246273 considered as a permutation of natural numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 5, 15, 11, 21, 16, 28, 9, 36, 22, 45, 29, 55, 20, 66, 37, 78, 8, 91, 14, 105, 46, 120, 56, 136, 35, 153, 13, 171, 67, 190, 77, 210, 79, 231, 92, 253, 27, 276, 106, 300, 12, 325, 104, 351, 121, 378, 26, 406, 170, 435, 137, 465, 154, 496, 65, 528, 43, 561, 172, 595, 209, 630, 191, 666, 211, 703, 54, 741, 18

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A246273.
Related permutations: A209268, A246675, A246276.
Cf. A055396, A246277.

(define (A246274 n) (let ((x (A246277 (+ 1 n))) (y (A055396 (+ 1 n)))) (* (/ 1 2) (- (expt (+ x y) 2) x y y y -2))))

a(n) = ((x+y)^2 - x - 3y + 2)/2, where x = A246277(n+1) and y = A055396(n+1).
As a composition of related permutations:
a(n) = A209268(A246675(n)).

A207800 Permutation of positive numbers. See comments.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 5, 16, 15, 32, 6, 64, 11, 128, 9, 256, 31, 512, 10, 1024, 13, 2048, 12, 4096, 23, 8192, 17, 16384, 14, 32768, 18, 65536, 63, 131072, 20, 262144, 19, 524288, 24, 1048576, 27, 2097152, 33, 4194304, 21, 8388608, 34, 16777216, 47, 33554432, 36, 67108864, 22, 134217728, 40

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

nonn

a(1)=1; on places 2,4,6,8,... we put 2^m, m=1,2,3,..., i.e., numbers n with A000120(n)=1; on places 3,7,11,15,... we put numbers n with A000120(n)=2; on places 5,13,21,29,... we put numbers n with A000120(n)=3; etc.

For general description of the order, see comment in A207790.

Cf. A207790.

a(n) = A066884(A209268(A065190(n))). Equivalently, a(n) = A067576(A249725(A065190(n))). - Ivan Neretin, Apr 30 2016

a(30) corrected by Ivan Neretin, Apr 30 2016

A207801 Permutation of positive numbers. See comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 14, 21, 16, 11, 18, 27, 20, 35, 22, 33, 24, 49, 26, 39, 28, 55, 30, 45, 32, 13, 34, 51, 36, 65, 38, 57, 40, 77, 42, 63, 44, 85, 46, 69, 48, 121, 50, 75, 52, 95, 54, 81, 56, 91, 58, 87, 60, 115, 62, 93, 64, 17, 66, 99, 68, 125, 70, 105, 72, 119, 74

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

nonn

a(1)=1; on places 2,4,6,8,... we put the same numbers, i.e., numbers with the least prime divisor p_1=2; on places 3,7,11,15,... we put numbers with the least prime divisor p_2=3; on places 5,13,21,29,... we put numbers with the least prime divisor p_3=5; etc.

For general description of the order, see comment in A207790.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A207790, A207800.

a = Array[1 &, mx = 74]; cnt = mx - 1; offs = Table[2^(i - 1) + 1, {i, 1, mx}]; n = 1; While[cnt > 0, n++; idx = PrimePi[FactorInteger[n][[1, 1]]]; pos = offs[[idx]]; If[pos > mx, Continue[]]; offs[[idx]] += 2^idx; a[[pos]] = n; cnt--]; a (* Ivan Neretin, May 06 2015 *)

For n>1, a(n) = A083140(1+A209268(n-1)). Equivalently, a(n) = A083221(1+A249725(n-1)). - Ivan Neretin, Apr 30 2016

A250252 Inverse permutation to A114881.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 5, 15, 11, 21, 16, 28, 9, 36, 22, 45, 29, 55, 14, 66, 37, 78, 8, 91, 20, 105, 46, 120, 56, 136, 27, 153, 13, 171, 67, 190, 35, 210, 79, 231, 92, 253, 44, 276, 106, 300, 12, 325, 54, 351, 121, 378, 19, 406, 65, 435, 137, 465, 154, 496, 77, 528, 26, 561, 172, 595, 90, 630, 191, 666, 211, 703, 104, 741, 18

Offset: 1

Antti Karttunen, Nov 15 2014

nonn

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

Inverse: A114881.
Similar or related permutations: A209268, A249812.
Cf. A000124, A000217, A005408, A006093, A055396, A078898.
Differs from A246274 for the first time at n=20, where a(20) = 14, while
A246274(20) = 20.

(define (A250252 n) (let ((x (A078898 (+ 1 n))) (y (A055396 (+ 1 n)))) (* (/ 1 2) (- (expt (+ x y) 2) x y y y -2))))

a(n) = 1 + ((((x+y)^2) - x - 3*y)/2), where x = A078898(n+1) and y = A055396(n+1).
As a composition of related permutations:
a(n) = A209268(A249812(n)).
Other identities. For all n >= 1 the following holds:
a(A005408(n-1)) = A000217(n). [Maps odd numbers to triangular numbers.]
a(A006093(n)) = A000124(n-1). [Maps precedents of primes to central polygonal numbers.]

A207802 Permutation of positive numbers. See comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 16, 7, 32, 9, 64, 10, 128, 12, 256, 11, 512, 18, 1024, 15, 2048, 24, 4096, 14, 8192, 27, 16384, 20, 32768, 36, 65536, 13, 131072, 48, 262144, 25, 524288, 54, 1048576, 21, 2097152, 72, 4194304, 30, 8388608, 81, 16777216, 22, 33554432, 96, 67108864, 40, 134217728, 108

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

nonn

a(1)=1; on places 2,4,6,8,... we put 2^m, m=1,2,3,..., i.e., numbers with the greatest prime divisor p_1=2; on places 3,7,11,15,... we put numbers with the greatest prime divisor p_2=3; on places 5,13,21,29,... we put numbers with the greatest prime divisor p_3=5; etc.

For general description of the order, see the comment in A207790.

Cf. A207790, A207800, A207801.

For n>1, a(n) = A125624(A209268(n-1)). - Ivan Neretin, Apr 30 2016

a(30) fixed by Ivan Neretin, Apr 30 2016

cp's OEIS Frontend

A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

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A246676 Permutation of natural numbers: a(n) = A242378(A007814(n), (1+A000265(n))) - 1.

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A249725 Inverse permutation to A135764.

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A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

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A246274 Inverse of A246273 considered as a permutation of natural numbers.

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A207800 Permutation of positive numbers. See comments.

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A207801 Permutation of positive numbers. See comments.

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A250252 Inverse permutation to A114881.

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