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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.

%I A331027 #30 Apr 13 2020 01:39:52
%S A331027 1,2,4,8,10,32,74,90,136,290,360,388,394,674,802,872,1184,1394,3316,
%T A331027 4100,5300,5634,10728,23098,25128,26836,33508,53954,61092,66610,92858,
%U A331027 187540,190120,215104,217732,955620
%N 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.
%C A331027 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.
%H A331027 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ulam_spiral">Ulam Spiral</a>.
%e A331027 The below table shows the details of the record step lengths of this sequence. The coordinate is relative to the starting 1-square.
%e A331027 --------------------------------------------------------------------------------
%e A331027     a(n)  | A330979 step # |   Start prime & coord   |  End prime & coord      |
%e A331027 --------------------------------------------------------------------------------
%e A331027        1  |         1      |         1 (0,0)         |         2 (1,0)         |
%e A331027        2  |         3      |         3 (1,1)         |        11 (2,0)         |
%e A331027        4  |         8      |        59 (2,4)         |        61 (0,4)         |
%e A331027        8  |        14      |       193 (-3,7)        |       101 (-5,5)        |
%e A331027       10  |        38      |       167 (4,-6)        |        83 (5,-3)        |
%e A331027       32  |        59      |       631 (13,7)        |      1103 (17,3)        |
%e A331027       74  |       169      |       113 (-3,-5)       |        53 (4,0)         |
%e A331027       90  |       319      |     17239 (66,12)       |     22291 (75,15)       |
%e A331027      136  |      1152      |      2719 (-26,12)      |      4127 (-32,2)       |
%e A331027      290  |      1659      |     13187 (19,-57)      |      7907 (30,-44)      |
%e A331027      360  |      2607      |     45263 (0,-106)      |     40283 (-18,-100)    |
%e A331027      388  |      7397      |     29723 (-86,-52)     |     35509 (-94,-70)     |
%e A331027      394  |      7806      |     47653 (-109,-19)    |     59663 (-122,-4)     |
%e A331027      674  |      7877      |     83101 (-144,-12)    |    114419 (-169,-5)     |
%e A331027      802  |     24920      |   2637497 (-812,692)    |   2515477 (-793,713)    |
%e A331027      872  |     27038      |   1285799 (-409,567)    |   1170607 (-423,541)    |
%e A331027     1184  |     55427      |    720089 (-288,-424)   |    653761 (-316,-404)   |
%e A331027     1394  |     56478      |    460349 (-339,-325)   |    457687 (-304,-338)   |
%e A331027     3316  |     56480      |    452293 (-300,-336)   |    410203 (-320,-282)   |
%e A331027     4100  |     82533      |    156353 (198,130)     |    129263 (158,180)     |
%e A331027     5300  |     83192      |    394211 (-140,314)    |    331697 (-208,288)    |
%e A331027     5634  |    165879      |     63589 (-126,42)     |    161761 (-201,45)     |
%e A331027    10728  |    237806      |   1034387 (509,-411)    |    962543 (491,-309)    |
%e A331027    23098  |    556765      |    110603 (-120,-166)   |     19249 (-3,-69)      |
%e A331027    25128  |    770967      |   7070333 (1330,-1278)  |   8614337 (1468,-1356)  |
%e A331027    26836  |   1074758      |   3213377 (-576,-896)   |   3582083 (-420,-946)   |
%e A331027    33508  |   1074809      |   4140079 (-129,-1017)  |   2995469 (-27,-865)    |
%e A331027    53954  |   2257389      |  67480409 (-1709,-4107) |  72669481 (-1882,-4262) |
%e A331027    61092  |   2644510      |   5269679 (790,1148)    |   5492621 (544,1172)    |
%e A331027    66610  |   2644988      |   1156873 (366,538)     |    694591 (417,285)     |
%e A331027    92858  |   2669627      |    109789 (166,62)      |    122443 (-117,175)    |
%e A331027   187540  |   2730402      |   2509621 (-792,228)    |    672787 (-410,24)     |
%e A331027   190120  |   2730411      |    193771 (-220,50)     |    296827 (74,-272)     |
%e A331027   215104  |   2730444      |   1505201 (285,-613)    |   4506473 (405,-1061)   |
%e A331027   217732  |   8160823      |  61908241 (-3934,1118)  |  48110423 (-3468,1142)  |
%e A331027   955620  |   8165267      |  22147771 (-2353,1019)  |   8236981 (-1435,1355)  |
%Y A331027 Cf. A000040, A330979, A063826, A214664, A214665, A136626, A115258.
%K A331027 nonn,walk,more
%O A331027 1,2
%A A331027 _Scott R. Shannon_, Jan 07 2020
%E A331027 a(17)-a(20) from _Scott R. Shannon_, Jan 10 2020
%E A331027 a(7) and beyond corrected by _Scott R. Shannon_, Feb 02 2020