A246275 -id:A246275 - cp's OEIS frontend

A246276 Inverse of A246275 considered as a permutation of natural numbers.

1, 3, 2, 6, 4, 10, 7, 5, 11, 15, 16, 21, 22, 8, 29, 28, 37, 36, 46, 17, 56, 45, 67, 9, 79, 12, 92, 55, 106, 66, 121, 30, 137, 13, 154, 78, 172, 68, 191, 91, 211, 105, 232, 23, 254, 120, 277, 14, 301, 93, 326, 136, 352, 24, 379, 155, 407, 153, 436, 171, 466, 57, 497, 39, 529, 190, 562, 192, 596, 210, 631, 231, 667, 47, 704, 19

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: A246275.

Cf. A055396, A246277, A246274, A246675.

(define (A246276 n) (let ((x (A055396 (+ 1 n))) (y (A246277 (+ 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 = A055396(n+1) and y = A246277(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)).

A250243 Permutation of natural numbers: a(n) = A246275(A055396(n+1), a(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, 51, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 39, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 87, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 63, 76, 101, 78, 79, 98, 81

Offset: 1

Antti Karttunen, Nov 16 2014

nonn

This is a "more recursed" variant of A249816. Preserves the parity of n.

Antti Karttunen, Table of n, a(n) for n = 1..5749
Index entries for sequences that are permutations of the natural numbers

Inverse: A250244.
Cf. A006093, A055396, A078898, A246275, A246278.
Similar or related permutations: A246684, A249813, A250246.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from "shallow variant" A249816 for the first time at n=39, where a(39) = 51, while A249816(39) = 39.

a(n) = A246275(A055396(n+1), a(A078898(n+1))).
As a composition of other permutations:
a(n) = A246684(A249813(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A006093(n)) = A006093(n). [Primes minus one are among the fixed points].

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].

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

Original entry on oeis.org

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

Offset: 2

Antti Karttunen, Aug 21 2014

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
This array can be obtained by taking every second column from array A242378, starting from its column 2.
Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n).
Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1.
From Antti Karttunen, Jan 03 2015: (Start)
A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.
(End)

			The top left corner of the array:
   2,     4,     6,     8,    10,    12,    14,    16,    18, ...
   3,     9,    15,    27,    21,    45,    33,    81,    75, ...
   5,    25,    35,   125,    55,   175,    65,   625,   245, ...
   7,    49,    77,   343,    91,   539,   119,  2401,   847, ...
  11,   121,   143,  1331,   187,  1573,   209, 14641,  1859, ...
  13,   169,   221,  2197,   247,  2873,   299, 28561,  3757, ...

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of the array

First row: A005843 (the even numbers), from 2 onward.
Row 2: A249734, Row 3: A249827.
Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
Transpose: A246279.
Inverse permutation: A252752.
One more than A246275.
Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 (A078898), A253551 (* A156552), A253561 (* A122111), A341605 (A017665), A341606 (A017666), A341607 (A006530 o A017666), A341608 (A341524), A341626 (A341526), A341627 (A341527), A341628 (A006530 o A341527), A342674 (A341530), A344027 (* A003415, arithmetic derivative), A355924 (A342671), A355925 (A009194), A355926 (A355442), A355927 (* sigma), A356155 (* A258851), A372562 (A252748), A372563 (A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
Cf. A329050 (subtable).

f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 *)

(define (A246278 n) (if (<= n 1) n (A246278bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here.
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
As a composition of other similar sequences:
a(n) = A122111(A253561(n)).
a(n) = A249818(A083221(n)).
For all n >= 1, a(n+1) = A005940(1+A253551(n)).
A(n, k) = A341606(n, k) * A355925(n, k). - Antti Karttunen, Jul 22 2022

Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015

A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(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, 18, 21, 256, 23, 12, 25, 14, 27, 512, 29, 1024, 31, 26, 33, 20, 35, 2048, 37, 42, 39, 4096, 41, 8192, 43, 22, 45, 16384, 47, 24, 49, 50, 51, 32768, 53, 36, 55, 66, 57, 65536, 59, 131072, 61, 38, 63, 52, 65, 262144, 67, 74, 69

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

Consider the square array A246278, and also A246275 which is obtained from the former when one is subtracted from each term.
In A246278 the even numbers occur at the top row, and all the rows below that contain only odd numbers, those subsequent terms in each column having been obtained by shifting all primes present in the prime factorization of number immediately above to one larger indices with A003961.
To compute a(n): we do the same process in reverse, by shifting primes in the prime factorization of n+1 step by step to smaller primes, until after k >= 0 such shifts with A064989, the result is even, with the smallest prime present being 2.
We subtract one from this even number and shift the binary expansion of the resulting odd number k positions left (i.e. multiply it with 2^k), which will be the result of a(n).
In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the 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.
A055396(n+1) tells on which row of A246275 n is, which is equal to the row of A246278 on which n+1 is.
A246277(n+1) tells in which column of A246275 n is, which is equal to the column of A246278 in which n+1 is.

			Consider 54 = 55-1. To find 55's position in array A246278, we start shifting its prime factorization 55 = 5 * 11 = p_3 * p_5, step by step: p_2 * p_4 (= 3 * 7 = 21), until we get an even number: p_1 * p_3 = 2*5 = 10.
This tells us that 55 is on row 3 and column 5 (= 10/2) of array A246278, thus 54 occurs in the same position at array A246275. In array A135764 the same position contains number (2^(3-1)) * (10-1) = 4*9 = 36, thus a(54) = 36.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246676.
More recursed variants: A246677, A246683.
Even bisection halved: A246679.
Other related permutations: A054582, A135764, A246274, A246275, A246276.
Cf. A000079, A003961, A005408, A055396, A064989, A246277, A246278.
a(n) differs from A156552(n+1) for the first time at n=13, where a(13) = 14, while A156552(14) = 17.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246675(n) = { my(k=0); n++; while((n%2), n = A064989(n); k++); n--; while(k>0, n = 2*n; k--); n; };
for(n=1, 2048, write("b246675.txt", n, " ", A246675(n)));

(define (A246675 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (-1+ (* 2 (A246277 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1)  * ((2*A246277(n+1))-1).
As a composition of related permutations:
a(n) = A135764(A246276(n)).
a(n) = A054582(A246274(n)-1).
Other identities. For all n >= 0:
a(A005408(n)) = A005408(n). [Fixes the odd numbers.]

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].

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].

cp's OEIS Frontend

A246276 Inverse of A246275 considered as a permutation of natural numbers.

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A246273 Transpose of square array A246275.

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A250243 Permutation of natural numbers: a(n) = A246275(A055396(n+1), a(A078898(n+1))).

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

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A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

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A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

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

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

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