A257340 -id:A257340 - cp's OEIS frontend

A257456 Smallest m, such that A257340(m) = n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 12, 16, 15, 18, 14, 20, 17, 24, 23, 22, 19, 28, 21, 26, 25, 33, 27, 40, 29, 31, 35, 36, 30, 42, 32, 44, 37, 38, 34, 50, 39, 46, 45, 48, 41, 52, 43, 54, 47, 57, 49, 61, 51, 64, 55, 59, 53, 68, 56, 66, 58, 70, 60, 72, 62

Offset: 1

Reinhard Zumkeller, Apr 24 2015

nonn

A257340(a(n)) = n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257340.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a257456 = (+ 1) . fromJust . (`elemIndex` a257340_list)

A257321 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "2" arm.

Original entry on oeis.org

2, 5, 7, 5, 3, 11, 5, 2, 7, 2, 3, 2, 5, 2, 5, 7, 3, 5, 7, 2, 5, 7, 11, 3, 7, 3, 5, 3, 5, 3, 5, 3, 7, 3, 2, 5, 2, 7, 2, 3, 5, 7, 2, 7, 2, 7, 2, 7, 2, 5, 11, 2, 5, 2, 3, 2, 5, 2, 5, 7, 3, 7, 3, 7, 3, 5, 3, 5, 3, 5, 7, 2, 5, 7, 3, 7, 3, 7, 3, 2, 5, 2, 7, 2, 3, 5

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Place numbers 2,3,5,7 clockwise around a grid point (see illustrations in links). Divide grid into four spiral arms.
Extend each arm one step at a time, in rotation: first the 2 arm, then the 3 arm, then the 5 arm,  then the 7 arm, then the 2 arm, etc.
Rule for extending: next term in arm is smallest number such that each cell in the grid is relatively prime to its eight neighbors.
Repetitions in arms are permitted.
The four arms are A257321, A257322, A257323, A257324.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Problem 146: Gcd, Vol. 4 (No. 45, Dec 1976), page PC45-4.
N. J. A. Sloane, Spirals showing initial terms of A257321-A257332

Cf. A064413, A257322-A257340.

More terms from Lars Blomberg, Apr 27 2015

A257332 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "7" arm.

Original entry on oeis.org

7, 9, 23, 15, 14, 47, 12, 59, 22, 65, 83, 26, 97, 101, 103, 119, 81, 137, 133, 143, 149, 93, 161, 48, 179, 42, 191, 54, 199, 129, 217, 66, 233, 56, 247, 82, 257, 271, 88, 283, 92, 183, 98, 305, 104, 319, 325, 329, 335, 231, 353, 355, 373, 377, 118, 383, 401

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Place numbers 2,3,5,7 clockwise around a grid point (see illustrations in links). Divide grid into four spiral arms.
Extend each arm one step at a time, in rotation: first the 2 arm, then the 3 arm, then the 5 arm, then the 7 arm, then the 2 arm, etc.
Rule for extending: next term in arm is smallest number such that each cell in the grid is relatively prime to its eight neighbors. Every term in the entire grid must be different.
The four arms are A257329, A257330, A257331, A257332.
Conjecture: every number > 1 appears in one of the four arms.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Problem 146: Gcd, Vol. 4 (No. 45, Dec 1976), page PC45-4.
N. J. A. Sloane, Spirals showing initial terms of A257321-A257332

Cf. A064413, A257321-A257340, A257347 (the union list).

More terms from Lars Blomberg, Apr 27 2015

A257339 Arrange numbers in a single clockwise spiral so that each number is relatively prime to its eight neighbors.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 9, 13, 17, 19, 8, 23, 29, 21, 25, 31, 37, 41, 35, 43, 47, 10, 49, 12, 53, 14, 15, 59, 27, 55, 39, 16, 61, 22, 51, 67, 18, 65, 24, 71, 26, 33, 32, 57, 73, 79, 83, 77, 85, 89, 97, 101, 28, 95, 34, 91, 38, 103, 107, 45, 109, 113, 20, 121

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Start with 1; always choose smallest number which has not yet appeared.

It is conjectured that every number appears.

Jon E. Schoenfield and Reinhard Zumkeller, Table of n, a(n) for n = 1..100000 (First 10000 terms from Jon E. Schoenfield)
Jon E. Schoenfield, Illustration of initial terms
Jon E. Schoenfield, Graph of first 18000 terms

Cf. A064413, A257321-A257340.
Cf. A257111 (first differences), A257455 (putative inverse).
Indices of primes and prime powers: A257457 and A257458.

Corrected and extended by Jon E. Schoenfield, Apr 23 2015

A257112 Arrange numbers in a clockwise spiral with initial terms a(1)=1, a(2)=2, a(4)=4, a(6)=6, a(8)=8, a(11)=3, a(15)=5, a(19)=7, a(23)=9; thereafter each number is relatively prime to all of its four (N,S,E,W) neighbors, but shares a factor with each of its (N,S,E,W) neighbors at distance 2 and also satisfies an additional condition stated in the comments.

Original entry on oeis.org

1, 2, 11, 4, 55, 6, 25, 8, 165, 14, 3, 16, 15, 26, 5, 12, 35, 18, 7, 22, 21, 32, 9, 28, 27, 10, 33, 20, 77, 34, 49, 38, 231, 46, 121, 24, 143, 36, 65, 44, 45, 52, 51, 58, 75, 56, 39, 40, 57, 50, 63, 62, 69, 64, 81, 68, 87, 17, 93, 136, 105, 74, 85, 42, 95, 48, 115, 54, 161

Offset: 1

Vladimir Shevelev, Apr 24 2015

nonn

To formulate the additional condition, let us call two numbers strictly connected if the set of prime divisors of one of them is a subset of the set of prime divisors of the other. Then the positions of two strictly connected terms should not be a knight's move apart.
Start with smallest number which has not yet appeared and satisfies the conditions: a(3)=11; thereafter always choose smallest number which has not yet appeared and satisfies the conditions.
This is a two-dimensional spiral analog of A098550.
In A098550 we have initial terms in the positions 1,2,3.
In the two-dimensional case we have 4 sides. So the initial TERMS are
      9
      8
  7 6 1 2 3     (1)
      4
      5
But the POSITIONS in the spiral are indexed thus:
.
   7--8--9--10
   |
   6  1--2
   |     |
   5--4--3
.
So the initial terms, by (1), are a(1)=1, a(2)=2, a(4)=4, a(6)=6, a(8)=8, ...
Conjecture: the sequence is a permutation of the positive integers. - Vladimir Shevelev, May 06 2015

			The spiral begins
.
   21---32----9---28---27---10  etc.
    |
   22   25----8--165---14
    |    |              |
    7    6    1----2    3
    |    |         |    |
   18   55----4---11   16
    |                   |
   35---12----5---26---15
.
Formally the smallest a(12) is 10, but then 10 and 5 are strictly connected numbers on a knight move (and a(13) would not exist). So the smallest suitable a(12)=16.

Peter J. C. Moses, Table of n, a(n) for n = 1..5625
Peter J. C. Moses, The first few squares.

Cf. A098550, A257321-A257340, A255370.

More terms from Peter J. C. Moses, Apr 29 2015

A253279 Arrange numbers in a clockwise spiral with initial terms a(1)=1, a(2)=2, a(4)=3, a(6)=4, a(8)=5; thereafter each number shares a factor with each of its four (N,S,E,W) neighbors.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 10, 5, 20, 8, 14, 16, 18, 9, 15, 21, 7, 28, 22, 24, 26, 30, 25, 35, 40, 32, 34, 36, 38, 42, 27, 33, 39, 45, 48, 56, 44, 70, 46, 50, 52, 13, 65, 60, 51, 75, 55, 66, 54, 57, 72, 58, 62, 64, 68, 78, 63, 69, 81, 84, 80, 74, 76, 82, 41

Offset: 1

Vladimir Shevelev, May 02 2015

nonn

Start with smallest number which has not yet appeared and satisfies the conditions: a(3)=6; thereafter always choose smallest number which has not yet appeared and satisfies the conditions.
This is a two-dimensional spiral analog of EKG sequence A064413.
In A064413 we have initial terms in the positions 1,2.
In the two-dimensional case we have 4 sides.
So the initial TERMS are
     5
   4 1 2      (1)
     3
But the POSITIONS in the spiral are indexed thus:
.
   7--8--9--10
   |
   6  1--2
   |     |
   5--4--3
.
So the initial terms, by (1), are a(1)=1, a(2)=2, a(4)=3, a(6)=4, a(8)=5.
Conjecture: The sequence is a permutation of the positive integers. - Vladimir Shevelev, May 06 2015

			The spiral begins
.
   26--30--25--35--40--32  etc.
    |
   24  10---5--20---8
    |   |           |
   22   4   1---2  14
    |   |       |   |
   28  12---3---6  16
    |               |
    7--21--15---9--18

Peter J. C. Moses, Table of n, a(n) for n = 1..5625
Peter J. C. Moses, The first few squares.

Cf. A064413, A257321-A257340, A257112.

Correction of a(42) and more terms from Peter J. C. Moses, May 03 2015

A257329 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "2" arm.

Original entry on oeis.org

2, 11, 13, 25, 21, 37, 27, 10, 39, 20, 71, 24, 85, 32, 95, 107, 115, 121, 125, 46, 145, 151, 155, 99, 167, 105, 181, 117, 197, 205, 211, 141, 223, 147, 76, 159, 86, 263, 72, 259, 135, 289, 30, 311, 60, 301, 94, 337, 116, 341, 343, 110, 359, 112, 237, 122, 389

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Place numbers 2,3,5,7 clockwise around a grid point (see illustrations in links). Divide grid into four spiral arms.
Extend each arm one step at a time, in rotation: first the 2 arm, then the 3 arm, then the 5 arm, then the 7 arm, then the 2 arm, etc.
Rule for extending: next term in arm is smallest number such that each cell in the grid is relatively prime to its eight neighbors. Every term in the entire grid must be different.
The four arms are A257329, A257330, A257331, A257332.
Conjecture: every number > 1 appears in one of the four arms.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Problem 146: Gcd, Vol. 4 (No. 45, Dec 1976), page PC45-4.
N. J. A. Sloane, Spirals showing initial terms of A257321-A257332

Cf. A064413, A257321-A257340, A257347 (the union list).

More terms from Lars Blomberg, Apr 27 2015

A257330 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "3" arm.

Original entry on oeis.org

3, 4, 17, 29, 31, 41, 49, 33, 61, 18, 73, 51, 77, 57, 28, 109, 34, 127, 38, 87, 62, 157, 40, 163, 169, 175, 187, 193, 64, 209, 203, 221, 227, 239, 153, 245, 171, 269, 177, 281, 293, 299, 189, 313, 201, 106, 207, 70, 219, 100, 347, 84, 361, 96, 379, 243, 391

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Place numbers 2,3,5,7 clockwise around a grid point (see illustrations in links). Divide grid into four spiral arms.
Extend each arm one step at a time, in rotation: first the 2 arm, then the 3 arm, then the 5 arm, then the 7 arm, then the 2 arm, etc.
Rule for extending: next term in arm is smallest number such that each cell in the grid is relatively prime to its eight neighbors. Every term in the entire grid must be different.
The four arms are A257329, A257330, A257331, A257332.
Conjecture: every number > 1 appears in one of the four arms.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Problem 146: Gcd, Vol. 4 (No. 45, Dec 1976), page PC45-4.
N. J. A. Sloane, Spirals showing initial terms of A257321-A257332

Cf. A064413, A257321-A257340, A257347 (the union list).

More terms from Lars Blomberg, Apr 27 2015

A257331 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "5" arm.

Original entry on oeis.org

5, 6, 19, 8, 35, 43, 16, 53, 67, 45, 79, 55, 89, 91, 69, 113, 36, 131, 63, 139, 75, 44, 111, 50, 173, 52, 185, 58, 123, 68, 215, 74, 229, 235, 241, 251, 253, 265, 277, 275, 287, 80, 307, 295, 317, 165, 331, 213, 323, 78, 349, 195, 367, 225, 371, 365, 397, 249

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Place numbers 2,3,5,7 clockwise around a grid point (see illustrations in links). Divide grid into four spiral arms.
Extend each arm one step at a time, in rotation: first the 2 arm, then the 3 arm, then the 5 arm, then the 7 arm, then the 2 arm, etc.
Rule for extending: next term in arm is smallest number such that each cell in the grid is relatively prime to its eight neighbors. Every term in the entire grid must be different.
The four arms are A257329, A257330, A257331, A257332.
Conjecture: every number > 1 appears in one of the four arms.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Problem 146: Gcd, Vol. 4 (No. 45, Dec 1976), page PC45-4.
N. J. A. Sloane, Spirals showing initial terms of A257321-A257332

Cf. A064413, A257321-A257340, A257347 (the union list).

More terms from Lars Blomberg, Apr 27 2015

A257322 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "3" arm.

Original entry on oeis.org

3, 2, 7, 3, 5, 7, 3, 7, 3, 2, 11, 3, 11, 3, 2, 5, 2, 7, 2, 7, 2, 11, 2, 5, 7, 3, 5, 7, 2, 11, 3, 5, 3, 5, 3, 5, 3, 7, 3, 7, 5, 7, 3, 7, 3, 2, 5, 2, 5, 2, 7, 2, 7, 2, 3, 11, 3, 2, 5, 2, 5, 2, 5, 7, 3, 5, 7, 2, 7, 2, 7, 2, 7, 2, 5, 11, 7, 3, 7, 3, 7, 3, 5, 3, 5

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Place numbers 2,3,5,7 clockwise around a grid point (see illustrations in links). Divide grid into four spiral arms.
Extend each arm one step at a time, in rotation: first the 2 arm, then the 3 arm, then the 5 arm, then the 7 arm, then the 2 arm, etc.
Rule for extending: next term in arm is smallest number such that each cell in the grid is relatively prime to its eight neighbors.
Repetitions in arms are permitted.
The four arms are A257321, A257322, A257323, A257324.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Problem 146: Gcd, Vol. 4 (No. 45, Dec 1976), page PC45-4.
N. J. A. Sloane, Spirals showing initial terms of A257321-A257332

Cf. A064413, A257321-A257340.

More terms from Lars Blomberg, Apr 27 2015

cp's OEIS Frontend

A257456 Smallest m, such that A257340(m) = n.

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Haskell

A257321 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "2" arm.

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A257332 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "7" arm.

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A257339 Arrange numbers in a single clockwise spiral so that each number is relatively prime to its eight neighbors.

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A253279 Arrange numbers in a clockwise spiral with initial terms a(1)=1, a(2)=2, a(4)=3, a(6)=4, a(8)=5; thereafter each number shares a factor with each of its four (N,S,E,W) neighbors.

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A257329 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "2" arm.

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A257330 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "3" arm.

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A257331 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "5" arm.

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A257322 Construct spiral of numbers on square grid as in Comments; sequence gives terms along the "3" arm.

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