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