A361374 Make a square spiral starting with a(1)=1, a(2)=2. Then, each position gets the smallest unused number which is the sum of a path of numbers starting from that position.

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

Offset: 1

Author

Samuel Harkness, Mar 28 2023

Keywords

• nonn

Comments

A path can go in any cardinal direction or diagonal. A path may not repeat the same number.

For a while, this sequence seems to simply be the natural numbers. However, the percentage of natural numbers in this sequence tends to 0. E.g., only 2347 of the first million natural numbers are in this sequence.

a(73) = 72 is the first to break from the natural numbers. 97 is the least positive number which does not occur.

Examples

			For a(42), the first candidate to check is 42, as it is the least unused positive integer. 20-22 is a valid path which ends at a(42) and whose sum is 42, so a(42) = 42. (path shown below)
.
   37   36   35   34   33   32   31
.
   38   17   16   15   14   13   30
.
   39   18    5    4    3   12   29
.
   40   19    6    1    2   11   28
.
   41   20    7    8    9   10   27
.     /    \
 start  21   22   23   24   25   26
.
For a(73), the first candidate to check is 73, as it is the least unused positive integer. No paths starting at a(73) equal 73, so check the next candidate, 74. 43-21-7-1-2 is a valid path starting at a(73) and whose sum is 74, so a(73) = 74. (path shown below)
.
   65   64   63   62   61   60   59   58   57
.
   66   37   36   35   34   33   32   31   56
.
   67   38   17   16   15   14   13   30   55
.
   68   39   18    5    4    3   12   29   54
.
   69   40   19    6    1----2   11   28   53
.                     /
   70   41   20    7    8    9   10   27   52
.               /
   71   42   21   22   23   24   25   26   51
.          /
   72   43   44   45   46   47   48   49   50
      /
 start
.
The first 144 terms:
.
  164-162-159-155-153-152-151-149-148-147-146-158
                                                |
  102-100--99--96--94--93--92--91--90--89-101 154
    |                                       |   |
  103  65--64--63--62--61--60--59--58--57  98 150
    |   |                               |   |   |
  104  66  37--36--35--34--33--32--31  56  95 141
    |   |   |                       |   |   |   |
  105  67  38  17--16--15--14--13  30  55  85 140
    |   |   |   |               |   |   |   |   |
  106  68  39  18   5---4---3  12  29  54  84 139
    |   |   |   |   |       |   |   |   |   |   |
  107  69  40  19   6   1---2  11  28  53  82 135
    |   |   |   |   |           |   |   |   |   |
  108  70  41  20   7---8---9--10  27  52  81 134
    |   |   |   |                   |   |   |   |
  110  71  42  21--22--23--24--25--26  51  88 133
    |   |   |                           |   |   |
  112  72  43--44--45--46--47--48--49--50  87 138
    |   |                                   |   |
  114  74--73--75--76--77--78--79--80--83--86 137
    |                                           |
  117-116-118-119-120-121-122-124-126-128-132-136
.
Note that 97 does not (and will not) occur. A path must start with one of the outer-most cells, all of which are greater than 97, and nothing below their minimum can ever be reached again.
		

Links

• Samuel Harkness, Table of n, a(n) for n = 1..10000
• Samuel Harkness, MATLAB program

Crossrefs

Cf. A174344, A274923 (spiral coordinates).

Cf. A141481, A334742.

Programs

MATLAB

See Links section.