A114881 -id:A114881 - cp's OEIS frontend

A250252 Inverse permutation to A114881.

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.]

A114882 Transposition sequence of A114881.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 8, 10, 9, 12, 11, 16, 24, 18, 13, 22, 15, 28, 48, 30, 17, 36, 14, 40, 120

Offset: 1

Clark Kimberling, Jan 03 2006

nonn

A self-inverse permutation of the natural numbers.

			Start with the northwest corner of A114881:
1 3 5 7 9
2 8 14 20 26
4 24 34 54 64
6 48 76 90 118
a(1)=1 because 1=T(1,1) and T(1,1)=1.
a(2)=3 because 2=T(2,1) and T(1,2)=3.
a(3)=2 because 3=T(1,2) and T(2,1)=2.
a(20)=48 because 20=T(2,4) and T(4,2)=48.

Cf. A083140, A114881.

(See A114538 for definition of transposition sequence.)

A249743 Main diagonal of square arrays A114881 and A249741.

Original entry on oeis.org

1, 8, 34, 90, 208, 376, 628, 816, 1218, 1768, 2200, 2922, 3648, 4342, 5028, 5988, 7728, 8478, 10116, 11572, 12628, 14298, 16018, 17710, 21630, 23128, 24616, 26856, 28666, 30622, 35686, 38382, 42606, 44062, 50212, 52698, 56362, 60798, 63960, 68680, 73210, 76200, 82702, 85498, 90028, 92136, 101068, 109492, 114180, 119308, 126052, 133122

Offset: 1

Antti Karttunen, Nov 15 2014

nonn

One less than the main diagonal of square arrays A083140 and A083221 formed from the sieve of Eratosthenes.

One less than A083141.

Cf. A000040, A083140, A083221, A114881, A249741.

(define (A249743 n) (if (= 1 n) n (- (* (A000040 n) (A000040 (+ n n -2))) 1)))

a(1) = 1, a(n) = (A000040(n) * A000040(2*(n-1))) - 1. [Where A000040(n) gives the n-th prime, p_n].
a(n) = A083140(n,n) - 1 = A083221(n,n) - 1.
a(n) = A083141(n+1)-1. [With the current starting offset 2 of A083141].

A249741 Sieve of Eratosthenes minus one: a(n) = A083221(n+1) - 1.

Original entry on oeis.org

1, 3, 2, 5, 8, 4, 7, 14, 24, 6, 9, 20, 34, 48, 10, 11, 26, 54, 76, 120, 12, 13, 32, 64, 90, 142, 168, 16, 15, 38, 84, 118, 186, 220, 288, 18, 17, 44, 94, 132, 208, 246, 322, 360, 22, 19, 50, 114, 160, 252, 298, 390, 436, 528, 28, 21, 56, 124, 202, 318, 376, 492, 550, 666, 840, 30, 23, 62, 144, 216, 340, 402, 526, 588, 712, 898, 960, 36, 25

Offset: 1

Antti Karttunen, Nov 15 2014

			The top left corner of the array:
   1,   3,   5,    7,    9,   11,   13,   15,   17,   19,   21,   23,   25,
   2,   8,  14,   20,   26,   32,   38,   44,   50,   56,   62,   68,   74,
   4,  24,  34,   54,   64,   84,   94,  114,  124,  144,  154,  174,  184,
   6,  48,  76,   90,  118,  132,  160,  202,  216,  258,  286,  300,  328,
  10, 120, 142,  186,  208,  252,  318,  340,  406,  450,  472,  516,  582,
  12, 168, 220,  246,  298,  376,  402,  480,  532,  558,  610,  688,  766,
  16, 288, 322,  390,  492,  526,  628,  696,  730,  798,  900, 1002, 1036,
  18, 360, 436,  550,  588,  702,  778,  816,  892, 1006, 1120, 1158, 1272,
  22, 528, 666,  712,  850,  942,  988, 1080, 1218, 1356, 1402, 1540, 1632,
  28, 840, 898, 1072, 1188, 1246, 1362, 1536, 1710, 1768, 1942, 2058, 2116,
...

Inverse: A249742.
Transpose: A114881.
Row 1: A005408, Column 1: A006093, Main diagonal: A249743.
Cf. also A002260, A004736, A038722, A083221.

(define (A249741 n) (- (A083221 (+ n 1)) 1))
;; Alternative version:
(define (A249741 n) (- (A083221bi (A002260 n) (A004736 n)) 1))

a(n) = A083221(n+1) - 1.
As a composition of related permutations:
a(n) = A114881(A038722(n)).
a(n) = A249811(A135764(n)).

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).

A249815 Permutation of natural numbers: a(n) = A249741(A055396(n+1), A246277(n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 21, 22, 23, 24, 25, 20, 27, 28, 29, 30, 31, 38, 33, 34, 35, 36, 37, 62, 39, 40, 41, 42, 43, 32, 45, 46, 47, 48, 49, 74, 51, 52, 53, 64, 55, 98, 57, 58, 59, 60, 61, 56, 63, 94, 65, 66, 67, 110, 69, 70, 71, 72, 73, 50, 75, 76, 77, 78, 79, 44, 81, 82, 83

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

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 A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.

Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A246273, as they are the transposes of above two arrays.

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

Inverse: A249816
Similar or related permutations: A250244 ("deep variant"), A246675, A249811, A249817, A246273, A246275, A114881, A249741.
Cf. A005408, A006093, A055396, A246277.
Differs from A249816 and A250243 for the first time at n=32, where a(32) = 38, while A249816(32) = A250243(32) = 44.
Differs from A250244 for the first time at n=39, where a(39) = 39, while A250244(39) = 51.

(define (A249815 n) (+ -1 (A083221bi (A246277 (+ 1 n)) (A055396 (+ 1 n))))) ;; Code for A083221bi given in A083221

a(n) = A249741(A055396(n+1), A246277(n+1)).
As a composition of other permutations:
a(n) = A249811(A246675(n)).
a(n) = A249817(n+1) - 1.
Other identities. For all n >= 1:
a(A005408(n-1)) = A005408(n-1) and a(A006093(n)) = A006093(n). [Fixes odd numbers and precedents of primes. Cf. comments above].

A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 8, 19, 9, 10, 23, 29, 12, 15, 31, 14, 37, 41, 16, 43, 25, 18, 47, 21, 20, 53, 59, 22, 27, 61, 24, 67, 71, 26, 35, 73, 28, 79, 33, 30, 83, 55, 32, 39, 89, 34, 97, 101, 36, 103, 107, 38, 109, 45, 40, 65, 49, 42, 51, 113, 44, 127, 85, 46, 131, 137, 48, 77, 57, 50, 139, 149, 52, 63, 151, 54, 95, 157, 56, 163, 121, 58, 167, 69, 60

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For n > 1, a(n) tells which number is located immediately above n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains 2n - 1.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A250471.
Odd bisection of A250470. The other bisection: A250479.
Cf. A055396, A064216, A078898, A083140, A083221, A114881, A114883.

a(1) = 1, a(n) = A083221(A055396(2*n - 1)-1, A078898(2*n - 1)).

a(n) = A250470(2*n - 1).

A246273 Transpose of square array A246275.

Original entry on oeis.org

1, 2, 3, 4, 8, 5, 6, 24, 14, 7, 10, 48, 34, 26, 9, 12, 120, 76, 124, 20, 11, 16, 168, 142, 342, 54, 44, 13, 18, 288, 220, 1330, 90, 174, 32, 15, 22, 360, 322, 2196, 186, 538, 64, 80, 17, 28, 528, 436, 4912, 246, 1572, 118, 624, 74, 19, 30, 840, 666, 6858, 390, 2872, 208, 2400, 244, 62, 21

Offset: 1

Antti Karttunen, Aug 21 2014

			The top-left corner of the array:
   1,     2,     4,     6,    10,    12,    16,    18,    22,   ...
   3,     8,    24,    48,   120,   168,   288,   360,   528,   ...
   5,    14,    34,    76,   142,   220,   322,   436,   666,   ...
   7,    26,   124,   342,  1330,  2196,  4912,  6858, 12166,   ...
   9,    20,    54,    90,   186,   246,   390,   550,   712,   ...
  11,    44,   174,   538,  1572,  2872,  5490,  8302, 15340,   ...
  ...

Inverse permutation: A246274.
Transpose: A246275.
Other related permutations: A038722, A054582, A246675, A246676.
One less than A246279.
Cf. A114881.

Scheme
```
(define (A246273 n) (- (A246279 n) 1))
```

a(n) = A246279(n) - 1.
As a composition of related permutations:
a(n) = A246275(A038722(n)).
a(n) = A246676(A054582(n-1)).

A249812 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 14, 21, 256, 23, 12, 25, 18, 27, 512, 29, 1024, 31, 22, 33, 20, 35, 2048, 37, 26, 39, 4096, 41, 8192, 43, 30, 45, 16384, 47, 24, 49, 34, 51, 32768, 53, 28, 55, 38, 57, 65536, 59, 131072, 61, 42, 63, 36, 65, 262144, 67, 46, 69, 524288, 71, 1048576, 73, 50, 75, 40, 77, 2097152, 79, 54, 81, 4194304, 83, 44

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A249741, the sieve of Eratosthenes minus 1. 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 A054582 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A249811.
Cf. A000079, A005408, A006093, A055396, A078898.
Similar or related permutations: A249813 ("deep variant"), A246675, A249816, A054582, A114881, A250252, A135764, A249741, A249742.
Differs from A246675 for the first time at n=20, where a(20)=14, while A246675(20)=18.

(define (A249812 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (+ -1 (* 2 (A078898 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).
As a composition of related permutations:
a(n) = A054582(A250252(n)-1).
a(n) = A135764(A249742(n)).
a(n) = A246675(A249816(n)).
Other identities. For all n >= 1 the following holds:
a(A006093(n)) = A000079(n-1).

A249816 Permutation of natural numbers: a(n) = A246275(A055396(n+1), A078898(n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 21, 22, 23, 24, 25, 20, 27, 28, 29, 30, 31, 44, 33, 34, 35, 36, 37, 32, 39, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 51, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 63, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 75, 76, 77, 78, 79, 98, 81, 82, 83

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A246275 is at the same position where n is in array A249741, the sieve of Eratosthenes minus 1. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.

Equally: a(n) tells which number in array A246273 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

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

Inverse: A249815.
Cf. A005408, A006093, A055396, A078898, A246278.
Similar or related permutations: A250243 ("deep variant"), A246676, A249812, A249818, A246273, A246275, A114881, A249741.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from A250244 for the first time at n=39, where a(39) = 39, while A250243(39) = 51.

(define (A249816 n) (+ -1 (A246278bi (A055396 (+ 1 n)) (A078898 (+ 1 n)))))

a(n) = A246275(A055396(n+1), A078898(n+1)).
As a composition of other permutations:
a(n) = A246676(A249812(n)).
a(n) = A249818(n+1) - 1.
Other identities. For all n >= 1:
a(A005408(n-1)) = A005408(n-1) and a(A006093(n)) = A006093(n). [Fixes odd numbers and precedents of primes. Cf. comments above].

cp's OEIS Frontend

A250252 Inverse permutation to A114881.

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A114882 Transposition sequence of A114881.

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A249743 Main diagonal of square arrays A114881 and A249741.

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A249741 Sieve of Eratosthenes minus one: a(n) = A083221(n+1) - 1.

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

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A249815 Permutation of natural numbers: a(n) = A249741(A055396(n+1), A246277(n+1)).

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A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

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A246273 Transpose of square array A246275.

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A249812 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).

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