A331027 The records for distance squared for step lengths between adjacent primes in A330979, the visited primes for a walk stepping to the closest unvisited prime on the Ulam Spiral.

1, 2, 4, 8, 10, 32, 74, 90, 136, 290, 360, 388, 394, 674, 802, 872, 1184, 1394, 3316, 4100, 5300, 5634, 10728, 23098, 25128, 26836, 33508, 53954, 61092, 66610, 92858, 187540, 190120, 215104, 217732, 955620

Offset: 1

Author

Scott R. Shannon, Jan 07 2020

Keywords

• nonn
• walk
• more

Comments

The sequence A330979 gives the visited primes for a walk on the Ulam Spiral which starts at 1 and then steps to the square containing the closest unvisited prime number. This sequences lists the records for the square of the step distance between primes for that walk. For a walk of 10 million steps the largest square distance is 955620, approximately 977.6 units, which occurs between A330979(8165267) = 22147771, which has coordinates (-2353,1019) relative to the starting 1-square, to A330979(8165268) = 8236981 with coordinates (-1435,1355). See A330979 for an image of the walk. It is unknown if this is a finite or infinite sequence.

Examples

			The below table shows the details of the record step lengths of this sequence. The coordinate is relative to the starting 1-square.
--------------------------------------------------------------------------------
    a(n)  | A330979 step # |   Start prime & coord   |  End prime & coord      |
--------------------------------------------------------------------------------
       1  |         1      |         1 (0,0)         |         2 (1,0)         |
       2  |         3      |         3 (1,1)         |        11 (2,0)         |
       4  |         8      |        59 (2,4)         |        61 (0,4)         |
       8  |        14      |       193 (-3,7)        |       101 (-5,5)        |
      10  |        38      |       167 (4,-6)        |        83 (5,-3)        |
      32  |        59      |       631 (13,7)        |      1103 (17,3)        |
      74  |       169      |       113 (-3,-5)       |        53 (4,0)         |
      90  |       319      |     17239 (66,12)       |     22291 (75,15)       |
     136  |      1152      |      2719 (-26,12)      |      4127 (-32,2)       |
     290  |      1659      |     13187 (19,-57)      |      7907 (30,-44)      |
     360  |      2607      |     45263 (0,-106)      |     40283 (-18,-100)    |
     388  |      7397      |     29723 (-86,-52)     |     35509 (-94,-70)     |
     394  |      7806      |     47653 (-109,-19)    |     59663 (-122,-4)     |
     674  |      7877      |     83101 (-144,-12)    |    114419 (-169,-5)     |
     802  |     24920      |   2637497 (-812,692)    |   2515477 (-793,713)    |
     872  |     27038      |   1285799 (-409,567)    |   1170607 (-423,541)    |
    1184  |     55427      |    720089 (-288,-424)   |    653761 (-316,-404)   |
    1394  |     56478      |    460349 (-339,-325)   |    457687 (-304,-338)   |
    3316  |     56480      |    452293 (-300,-336)   |    410203 (-320,-282)   |
    4100  |     82533      |    156353 (198,130)     |    129263 (158,180)     |
    5300  |     83192      |    394211 (-140,314)    |    331697 (-208,288)    |
    5634  |    165879      |     63589 (-126,42)     |    161761 (-201,45)     |
   10728  |    237806      |   1034387 (509,-411)    |    962543 (491,-309)    |
   23098  |    556765      |    110603 (-120,-166)   |     19249 (-3,-69)      |
   25128  |    770967      |   7070333 (1330,-1278)  |   8614337 (1468,-1356)  |
   26836  |   1074758      |   3213377 (-576,-896)   |   3582083 (-420,-946)   |
   33508  |   1074809      |   4140079 (-129,-1017)  |   2995469 (-27,-865)    |
   53954  |   2257389      |  67480409 (-1709,-4107) |  72669481 (-1882,-4262) |
   61092  |   2644510      |   5269679 (790,1148)    |   5492621 (544,1172)    |
   66610  |   2644988      |   1156873 (366,538)     |    694591 (417,285)     |
   92858  |   2669627      |    109789 (166,62)      |    122443 (-117,175)    |
  187540  |   2730402      |   2509621 (-792,228)    |    672787 (-410,24)     |
  190120  |   2730411      |    193771 (-220,50)     |    296827 (74,-272)     |
  215104  |   2730444      |   1505201 (285,-613)    |   4506473 (405,-1061)   |
  217732  |   8160823      |  61908241 (-3934,1118)  |  48110423 (-3468,1142)  |
  955620  |   8165267      |  22147771 (-2353,1019)  |   8236981 (-1435,1355)  |
		

Links

• Wikipedia, Ulam Spiral.

Crossrefs

Cf. A000040, A330979, A063826, A214664, A214665, A136626, A115258.

Extensions

a(17)-a(20) from Scott R. Shannon, Jan 10 2020

a(7) and beyond corrected by Scott R. Shannon, Feb 02 2020