A063826 -id:A063826 - cp's OEIS frontend

A347358 The prime numbers visited on a square spiral when starting at 1 and then stepping to the smallest unvisited prime number that is visible from the current number.

1, 2, 3, 11, 5, 13, 29, 17, 7, 19, 31, 23, 37, 53, 41, 61, 43, 59, 47, 71, 83, 67, 89, 73, 101, 79, 107, 127, 97, 131, 103, 137, 109, 139, 113, 149, 173, 151, 179, 157, 181, 163, 191, 167, 193, 227, 197, 229, 293, 233, 211, 239, 199, 251, 223, 257, 307, 241, 311, 263, 313, 269, 317, 271, 331, 277

Offset: 1

Scott R. Shannon, Aug 28 2021

nonn

A number is visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is 1. See A331400 for the points visible from the starting 1 number.

The primes visited in the sequence appear to oscillate between two different regimes. In one the vast majority of the next smallest visible primes are on the corners of the neighboring inner or outer square ring of numbers, thus the steps are nearly vertical or horizontal relative to the current square. In the other the majority of next smallest visible primes are on square rings much closer or further away from the origin than the current ring, or entirely on the other side of the spiral relative to the starting number. In this regime the path makes very random steps in many different diagonal directions, covering the entire spiral. See the three linked images.

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1. The central starting number.
a(2) = 2, a(3) = 3 as 2 is the smallest visible unvisited prime from 1, and 3 is the smallest visible unvisited prime from 2.
a(4) = 11 as 11 is the smallest visible unvisited prime from 3. Note that from 3 the smaller unvisited primes 5 and 7 are hidden from 3 by the numbers 4 and 1.
a(7) = 29 as 29 is the smallest visible unvisited prime from 13. Note that from 13 the smaller unvisited primes 7, 17, 19, 23 are hidden from 13 by numbers 3, 14, 4, 2 respectively.

Scott R. Shannon, Image of the path for the first 7000 terms. The colors are graduated across red, orange, yellow to show the relative step order. Note the yellow lines, terms in the 5000-7000 range, step in all directions across the entire spiral.
Scott R. Shannon, Image of the path for the first 14000 terms. The colors are now graduated across red, orange, yellow, green, blue. Note how the steps for the later colors, terms in the 1000-14000 range, are almost all horizontal or vertical and none step diagonally into the inner spiral.
Scott R. Shannon, Image of the path for the first 21000 terms. The colors are now graduated across red, orange, yellow, green, blue, indigo, violet. Note how the later colors, terms in the 15000-21000 range, again behave like the earlier 5000-7000 term range and step in random directions across the spiral.
Eric Weisstein's World of Mathematics, Visible Point.
Wikipedia, Ulam Spiral.

Cf. A347522 (step to smallest hidden), A000040, A063826, A214664, A214665, A331400, A335364, A332767, A330979.

A079421 Spiro-Fibonacci differences, a(n) = difference of two previous terms that are nearest when terms arranged in a spiral.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 0

Neil Fernandez, Jan 07 2003

nonn

			Terms are written in square boxes radiating spirally (cf. Ulam prime spiral). a(0)=0 and a(1)=1, so write 0 and then 1 to its right. a(2) goes in the box below a(1). The nearest two filled boxes contain a(0) and a(1), so a(2)=abs(a(0)-a(1))=abs(0-1)=1. a(3) goes in the box to the left of a(2). The nearest two filled boxes contain a(0) and a(2), so a(3)=abs(a(0)-a(2))=abs(0-1)=1.

Sean A. Irvine, Table of n, a(n) for n = 0..10000
N. Fernandez, Graphical representations of some Spiro-Fibonacci sequences

Cf. A063826, A078510.

A094768 Square spiral of sums of selected preceding terms, starting at 1 (a spiral Fibonacci-like sequence).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 42, 68, 110, 179, 291, 470, 763, 1236, 2005, 3241, 5252, 8502, 13770, 22272, 36058, 58355, 94455, 152878, 247333, 400279, 647722, 1048180, 1696193, 2744373, 4440857, 7185700, 11627320, 18814256, 30443581, 49257837

Offset: 1

Yasutoshi Kohmoto, Jun 10 2004

nonn

Enter 1 into center position of the spiral. Repeat: Add to the number in the present position the numbers in all those already filled positions that are horizontally or vertically adjacent to it, go to next position of the spiral and enter the sum into it.
a(1) = 1, a(n) = a(n-1) + Sum_{i < n-1 and a(i) is adjacent to a(n-1)} a(i).
Here only four positions are considered adjacent, eight however in A094767.
Clockwise and counterclockwise construction of the spiral result in the same sequence.

			Clockwise constructed spiral begins
.
  13770--22272--36058--58355--94455
      |
      |
   8502     16-----25-----42-----68
      |      |                    |
      |      |                    |
   5252      9      1------1    110
      |      |             |      |
      |      |             |      |
   3241      6------3------2    179
      |                           |
      |                           |
   2005---1236----763----470----291
.
where
  a(2) = a(1) = 1,
  a(3) = a(2) + a(1) = 2,
  a(4) = a(3) + a(2) = 3,
  a(5) = a(4) + a(3) + a(1) = 6,
  a(6) = a(5) + a(4) = 9,
  a(7) = a(6) + a(5) + a(1) = 16.

Klaus Brockhaus, Table of n, a(n) for n = 1..729

Cf. A063826, A094767, A094769, A126937, A141481.

{m=5; h=2*m-1; A=matrix(h, h); print1(A[m, m]=1, ","); pj=m; pk=m; T=[[1, 0], [0, -1], [ -1, 0], [0, 1]]; for(n=1, (h-2)^2-1, g=sqrtint(n); r=(g+g%2)\2; q=4*r^2; d=n-q; if(n<=q-2*r, j=d+3*r; k=r, if(n<=q, j=r; k=-d-r, if(n<=q+2*r, j=r-d; k=-r, j=-r; k=d-3*r))); j=j+m; k=k+m; s=A[pj, pk]; for(c=1, 4, v=[pj, pk]; v+=T[c]; s=s+A[v[1], v[2]]); A[j, k]=s; print1(s, ","); pj=j; pk=k)} \\ Klaus Brockhaus, Aug 27 2008

Edited and extended beyond a(14) by Klaus Brockhaus, Aug 27 2008

A214666 The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

Original entry on oeis.org

-1, -1, 1, 1, -2, -2, 2, 2, 0, -3, -3, 3, 3, 3, -1, -4, -2, 0, 4, 4, 4, -2, -5, -5, 1, 5, 5, 5, 5, 3, -6, -6, -2, 0, 6, 6, 6, 0, -4, -7, -7, -7, 1, 3, 7, 7, 7, -5, -8, -8, -8, -8, -8, 2, 8, 8, 8, 8, 4, 0, -2, -9, -9, -5, -3, 1, 9, 9, 5, 3, -1, -7, -10, -10, -10

Offset: 1

William Rex Marshall, Jul 26 2012

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This spiral is rotated 180 degrees from the spiral on the March 1964 cover of Scientific American.

William Rex Marshall, Table of n, a(n) for n = 1..10077 (coordinates of primes in a 325 X 325 square)

Cf. A000040, A063826, A214664, A214665, A214667.

A214667 The y-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

Original entry on oeis.org

0, -1, -1, 1, 0, -2, -2, 0, 2, -1, -3, -3, 1, 3, 3, 0, -4, -4, -2, 2, 4, 4, 3, -3, -5, -5, -3, 1, 3, 5, 0, -4, -6, -6, -2, 0, 6, 6, 6, 3, -3, -5, -7, -7, -7, -5, 7, 7, 6, 4, 0, -6, -8, -8, -8, -2, 4, 6, 8, 8, 8, 5, -9, -9, -9, -9, -3, 3, 9, 9, 9, 9, 4, -2, -8

Offset: 1

William Rex Marshall, Jul 26 2012

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This spiral is rotated 180 degrees from the spiral on the March 1964 cover of Scientific American.

William Rex Marshall, Table of n, a(n) for n = 1..10077 (coordinates of primes in a 325 X 325 square)

Cf. A000040, A063826, A214664, A214665, A214666.

A336402 Squares visited by a chess queen moving on a square-spiral numbered board where the queen moves to the closest unvisited square containing a prime number. In case of a tie it chooses the square with the smallest prime number.

Original entry on oeis.org

1, 2, 3, 11, 29, 13, 31, 59, 61, 97, 139, 191, 251, 193, 101, 103, 67, 37, 17, 5, 19, 7, 23, 47, 79, 163, 281, 353, 283, 433, 521, 617, 523, 619, 439, 359, 223, 167, 227, 293, 229, 173, 83, 233, 127, 53, 179, 131, 89, 137, 389, 313, 311, 467, 383, 307, 241, 239, 181, 457, 547, 643

Offset: 1

Scott R. Shannon, Jul 20 2020

This sequences gives the numbers of the squares visited by a chess queen moving on a square-spiral numbered board where the queen starts on the 1 numbered square and at each step moves to the closest unvisited square containing a prime number. The movement is restricted to the eight directions a queen can move on a standard chess board, and the queen cannot move over a previously visited square  If two or more unvisited prime numbered squares exist which are the same distance from the current square then the one with the smallest prime number is chosen. Note that if the queen simply moves to the closest unvisited square the sequence will be infinite as the queen will just follow the square spiral path.
The sequence is finite. After 519 steps the square with number 1289 is visited, after which all eight squares the queen can move to have been visited.
The first term where this sequence differs from A330979, which steps to the closest unvisited prime without any movement direction restrictions, is a(40) = 227. See the examples below.
The largest visited square is a(292) = 14843. The largest step distance between visited squares is 20 units, between a(338) = 2879 to a(339) = 3779. The largest prime gap between visited squares is 4050, from a(396) = 10667 to a(397) = 14717. The smallest unvisited prime is 41.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square for the queen.
a(2) = 2. The seven unvisited prime numbered squares around a(1) the queen can move to are numbered 2,3,61,5,19,7,23. Of these 2 is the closest, being 1 unit away. There are no primes in the south-east direction from a(1).
a(4) = 11. The four unvisited prime numbered squares around a(3) = 3 the queen can move to are numbered 11,29,13,5, the other two directions not having any primes. Both 11 and 13 are sqrt(2) units away, and of those 11 is the smallest.
a(40) = 227. The three unvisited prime numbered squares around a(39) = 167 the queen can move to are numbered 227,173,53, Of these 227 is the closest, being 4 units away. Note that the square with prime number 83 is only sqrt(10), about 3.16, units away but is at relative coordinates (1,3) to 167 so cannot be reach by the queen.

Scott R. Shannon, Image showing the 519 steps of the queen's path. A green square shows the starting 1 square, a red square shows the final square with number 1289, and a thick white line is the path between visited squares. All visited prime numbered squares are shown in yellow, while those unvisited squares containing primes are shown in grey. The eight squares which block the queen's movement from the final square are shown with a red border. The square spiral numbering of the board is shown as a thin white line. Click on the image to zoom in to see the prime numbers.

Cf. A336413, A336446, A336447, A330979, A000040, A063826, A214664, A214665, A136626, A115258, A331027.

A336446 Squares visited by a chess queen moving on a square-spiral numbered board where the queen moves to an unvisited square containing the smallest prime number.

Original entry on oeis.org

1, 2, 3, 5, 7, 19, 17, 13, 11, 23, 47, 43, 41, 37, 31, 29, 53, 127, 79, 73, 71, 67, 103, 101, 97, 61, 59, 131, 89, 83, 173, 167, 163, 157, 151, 107, 109, 271, 211, 199, 197, 193, 191, 139, 137, 239, 181, 179, 641, 457, 241, 251, 257, 263, 149, 397, 389, 313, 311, 307, 293, 113, 281

Offset: 1

Scott R. Shannon, Jul 22 2020

This sequences gives the numbers of the squares visited by a chess queen moving on a square-spiral numbered board where the queen starts on the 1 numbered square and at each step moves to an unvisited square containing the smallest prime number. The movement is restricted to the eight directions a queen can move on a standard chess board, and the queen cannot move over a previously visited square. Note that if the queen simply moves to an unvisited square containing the smallest number the sequence will be infinite as the queen will just follow the square spiral path.
The sequence is finite. After 5880 steps the square with number 55903 is visited, after which all eight squares the queen can move to have been visited.
The first term where this sequence differs from A336402, where the queen steps to the closest unvisited prime, is a(4) = 5. See the examples below.
The largest visited square is a(4943) = 79187. The largest step distance between visited squares is 72 units, between a(3205) = 31397 to a(3206) = 31469. The largest prime gap between visited squares is 30150, from a(4942) = 49037 to a(4943) = 79187. The smallest unvisited prime is 45833.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square for the queen.
a(2) = 2. The seven unvisited prime numbered squares around a(1) the queen can move to are numbered 2,3,61,5,19,7,23. Of these 2 is the smallest. There are no primes in the south-east direction from a(1).
a(4) = 5. The four unvisited prime numbered squares around a(3) = 3 the queen can move to are numbered 11,29,13,5, the other two available directions not having any primes. Of these 5 is the smallest. Note that 11 is the closest prime, being only sqrt(2) units away while 5 is 2 units away.
a(4943) = 79187. This is only unvisited square containing a prime number around a(4942) = 49037. It is 30 units away to the right.

Scott R. Shannon, Image showing the 5880 steps of the queen's path. A green square shows the starting 1 square, a red square, above the bottom-left corner, shows the final square with number 55903, and a thick white line is the path between visited squares. All visited prime numbered squares are shown in yellow, while those unvisited squares containing primes are shown in grey. The eight squares which block the queen's movement from the final square are shown with a red border. The square spiral numbering of the board is shown as a thin white line. Click on the image to zoom in to see the prime numbers.

Cf. A336402, A336413, A336447, A330979, A000040, A063826, A214664, A214665, A136626, A115258, A331027.

A344659 Lexicographically earliest sequence of distinct nonnegative terms on a square spiral such that each term forms no prime value in the eight sums when each term is added to each of its eight nearest neighbors.

Original entry on oeis.org

0, 1, 8, 14, 4, 6, 9, 15, 20, 5, 7, 13, 2, 12, 10, 11, 16, 22, 18, 3, 17, 19, 25, 29, 28, 21, 23, 27, 31, 26, 24, 30, 32, 34, 35, 33, 36, 41, 40, 42, 39, 37, 38, 43, 45, 47, 49, 53, 57, 48, 51, 54, 61, 50, 44, 46, 52, 56, 58, 60, 59, 65, 55, 62, 63, 70, 64, 74, 66, 69, 67, 68, 72, 73, 77, 71, 75

Offset: 1

Eric Angelini and Scott R. Shannon, May 26 2021

nonn

.
  36--33--35--34--32--30--24
   |                       |
  41  16--11--10--12---2  26
   |   |               |   |
  40  22   4--14---8  13  31
   |   |   |       |   |   |
  42  18   6   0---1   7  27
   |   |   |           |   |
  39   3   9--15--20---5  23   .
   |   |                   |   .
  37  17--19--25--29--28--21   .
   |                           |
  38--43--45--47--49--53--57--48
.

			The eight terms that are in contact with the initial zero are 1, 8, 14, 4, 6, 9, 15, 20: none of them is prime [forcing the sum a(k) + 0 to be nonprime, with k<9]; more generally, no term of the square spiral when added to any of its eight nearest neighbors sums to a prime.

Cf. A063826, A344660, A002808, A000040.

A335710 The smallest number on a square (Ulam) spiral in a 2D grid such that n steps in one of the four axial directions leads to each visited number sharing a common factor greater than 1 with the previous visited number.

Original entry on oeis.org

1, 3, 30, 1235, 2439, 90000, 88805, 4330458, 4322139, 22001763, 21983004, 1868098088, 2436807593

Offset: 0

Scott R. Shannon, Jun 18 2020

Start with any number on a square (Ulam) spiral in a 2D grid and then continue to step right to the next square as long as the number in that square shares a common factor > 1 with the number in the current square. Count the steps one can take. Repeat this process in each of the other three axial directions left, upward and downward, and then take the maximum step length of these four directions. The sequence a(n) gives the smallest number such that the maximum step length of these four directions is n.

If a(13) exists it is greater than 5*10^11.

			a(0) = 1 as 1 has no common factor > 1 with its neighboring four squares.
a(1) = 3 as stepping right one step from 3 leads to 12 which shares the common factor 3.
a(2) = 30 as stepping right two steps from 30 leads to 55 and 88 which share the common factors 5 and 11 respectively.
a(3) = 1235 as stepping right three steps from 1235 leads to 1380, 1533, 1694 which share the common factors 5, 3, 7 respectively.
a(4) = 2439 as stepping right four steps from 2439 leads to 2640, 2849, 3066, 3291 which share the common factors 3, 11, 7, 3 respectively.
a(5) = 90000 as stepping upward five steps from 90000 leads to 91203, 92414, 93633, 94860, 96095 which share common factors 3, 7, 23, 3, 5 respectively.
a(6) = 88805 as stepping upward one step from 88805 leads to 90000, which shares a common factor 5, and then continues upwards with the same five steps as a(5).
a(7) = 4330458 as stepping downward seven steps from 4330458 leads to 4338785, 4347120, 4355463, 4363814, 4372173, 4380540, 4388915 which share common factors 11, 5, 3, 7, 13, 3, 5 respectively.
a(8) = 4322139 as stepping downward one step from 4322139 leads to 4330458, which shares a common factor 3, and then continue downward with the same seven steps as a(7).
a(9) = 22001763 as stepping downward nine steps from 22001763 leads to 22020530, 22039305, 22058088, 22076879, 22095678, 22114485, 22133300, 22152123, 22170954 which share common factors 7, 5, 3, 19, 11, 3, 5, 7, 3 respectively.
a(10) = 21983004 as stepping downward one step from 21983004 leads to 22001763, which shares a common factor 3, and then continue downward with the same nine steps as a(9).
a(11) = 1868098088 as stepping upward eleven steps from 1868098088 leads to 1868270979, 1868443878, 1868616785, 1868789700, 1868962623, 1869135554, 1869308493, 1869481440, 1869654395, 1869827358, 1870000329 which share common factors 23, 3, 7, 5, 3, 11, 13, 3, 5, 7, 3 respectively.
a(12) = 2436807593 as stepping left twelve steps from 2436807593 leads to 2437005054, 2437202523, 2437400000, 2437597485, 2437794978, 2437992479, 2438189988, 2438387505, 2438585030, 2438782563, 2438980104, 2439177653 which share common factors 11, 3, 7, 5, 3, 23, 13, 3, 5, 7, 3, 11 respectively.

Cf. A000005, A063826, A214665, A330979, A332767, A335661.

A338642 Square spiral of smallest distinct positive integers starting at 1 such that the four sums of each term with its four nearest neighbors are composite numbers.

Original entry on oeis.org

1, 3, 5, 7, 2, 8, 4, 11, 9, 6, 12, 10, 14, 13, 15, 18, 16, 19, 17, 21, 23, 22, 24, 25, 20, 26, 28, 27, 29, 31, 32, 30, 33, 35, 34, 36, 38, 39, 37, 40, 41, 43, 42, 45, 46, 44, 47, 48, 50, 49, 51, 53, 55, 56, 54, 52, 58, 59, 57, 60, 61, 62, 63, 66, 64, 68, 65, 67, 71, 69, 72, 70, 73, 74, 76, 77, 75

Offset: 1

Eric Angelini and Scott R. Shannon, Apr 21 2021

nonn

			The square spiral starts:
.
     38--36--34--35--33--30--32
      |                       |
     39  16--18--15--13--14  31
      |   |               |   |
     37  19   2---7---5  10  29
      |   |   |       |   |   |
     40  17   8   1---3  12  27
      |   |   |           |   |
     41  21   4--11---9---6  28
      |   |                   |
     43  23--22--24--25--20--26
      |
     42--45--46--44--47--48--50..
.
a(2) = 3 as a(1) + 3 = 1 + 3 = 4, the smallest possible composite number.
a(3) = 5 as a(2) + 5 = 3 + 5 = 8. Note a(3) cannot be 2 or 4 as when these are added to 3 the result is a prime number.
a(4) = 7 as a(3) + 7 = 5 + 7 = 12, and a(1) + 7 = 1 + 7 = 8, both being composite.
a(9) = 9 as a(8) + 9 = 11 + 9 = 20, and a(2) + 9 = 3 + 9 = 12, both being composite.

Cf. A338644 (sum to primes), A002808, A063826, A260643, A334742, A307834, A338221.

cp's OEIS Frontend

A347358 The prime numbers visited on a square spiral when starting at 1 and then stepping to the smallest unvisited prime number that is visible from the current number.

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A079421 Spiro-Fibonacci differences, a(n) = difference of two previous terms that are nearest when terms arranged in a spiral.

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A094768 Square spiral of sums of selected preceding terms, starting at 1 (a spiral Fibonacci-like sequence).

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A214666 The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

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A214667 The y-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step west.

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A336402 Squares visited by a chess queen moving on a square-spiral numbered board where the queen moves to the closest unvisited square containing a prime number. In case of a tie it chooses the square with the smallest prime number.

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A336446 Squares visited by a chess queen moving on a square-spiral numbered board where the queen moves to an unvisited square containing the smallest prime number.

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A344659 Lexicographically earliest sequence of distinct nonnegative terms on a square spiral such that each term forms no prime value in the eight sums when each term is added to each of its eight nearest neighbors.

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A335710 The smallest number on a square (Ulam) spiral in a 2D grid such that n steps in one of the four axial directions leads to each visited number sharing a common factor greater than 1 with the previous visited number.

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A338642 Square spiral of smallest distinct positive integers starting at 1 such that the four sums of each term with its four nearest neighbors are composite numbers.

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