A246273 -id:A246273 - cp's OEIS frontend

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

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)).

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].

A246275 Square array A246278 minus 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 21 2014

			The top-left corner of the array:
   1,     3,     5,     7,     9,    11,    13,    15,    17,   ...
   2,     8,    14,    26,    20,    44,    32,    80,    74,   ...
   4,    24,    34,   124,    54,   174,    64,   624,   244,   ...
   6,    48,    76,   342,    90,   538,   118,  2400,   846,   ...
  10,   120,   142,  1330,   186,  1572,   208, 14640,  1858,   ...
  12,   168,   220,  2196,   246,  2872,   298, 28560,  3756,   ...
  ...

Inverse permutation: A246276.
Transpose: A246273.
One less than A246278.
Related permutations: A038722, A246675, A246676.
Cf. also A003961.

(define (A246275 n) (- (A246278 (+ 1 n)) 1))

a(n) = A246278(n+1)-1.
As a composition of related permutations:
a(n) = A246273(A038722(n)).

Formula edited slightly because of changed starting offset of A246278. - Antti Karttunen, Jan 27 2015

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].

A246279 Transpose of array A246278: A(row,1) = 2*row, and for col > 1, A(row,col) = A003961(A(row,col-1)).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 27, 10, 13, 121, 77, 125, 21, 12, 17, 169, 143, 343, 55, 45, 14, 19, 289, 221, 1331, 91, 175, 33, 16, 23, 361, 323, 2197, 187, 539, 65, 81, 18, 29, 529, 437, 4913, 247, 1573, 119, 625, 75, 20, 31, 841, 667, 6859, 391, 2873, 209, 2401, 245, 63, 22

Offset: 1

Antti Karttunen, Aug 21 2014

Permutation of natural numbers larger than 1.

See comments in A246278 (the same array transposed).

			The top-left corner of the array:
   2,     3,     5,     7,    11,    13,    17,    19,    23, ...
   4,     9,    25,    49,   121,   169,   289,   361,   529, ...
   6,    15,    35,    77,   143,   221,   323,   437,   667, ...
   8,    27,   125,   343,  1331,  2197,  4913,  6859, 12167, ...
  10,    21,    55,    91,   187,   247,   391,   551,   713, ...
  12,    45,   175,   539,  1573,  2873,  5491,  8303, 15341, ......

Transpose of A246278.
One more than A246273.
Cf. A003961, A083140.

(define (A246279 n) (A246278bi (A004736 n) (A002260 n)))
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

If col = 1, A(row,col) = 2*row, otherwise A(row,col) = A003961(A(row,col-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].

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

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

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A246275 Square array A246278 minus 1.

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

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A246279 Transpose of array A246278: A(row,1) = 2*row, and for col > 1, A(row,col) = A003961(A(row,col-1)).

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

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