

A336799


The numbers visited on a square spiral, with a(1)=1, a(2)=2, a(3) = 6, when stepping to an unvisited number as close as possible to the n = 1 starting position that has at least one common factor with the last visited number but none with the second last visited number. In case of a tie, choose the smallest number.


1



1, 2, 6, 15, 35, 14, 12, 33, 77, 28, 10, 45, 21, 56, 20, 55, 99, 18, 34, 85, 75, 24, 22, 143, 39, 30, 46, 161, 63, 36, 40, 95, 57, 42, 26, 65, 105, 48, 38, 247, 117, 69, 115, 70, 44, 187, 51, 54, 52, 91, 119, 68, 60, 87, 203, 98, 62, 93, 129, 86, 76, 133, 175, 50, 78, 141, 235, 80, 58, 261
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OFFSET

1,2


COMMENTS

This sequence is the square spiral version of the Enots Wolley sequence A336957. The same rules for selecting the next number apply except that, instead of choosing the smallest unvisited number for a(n), the number closest to the starting n = 1 position which satisfies the selection rules is chosen. If two or more such numbers exist then the smallest is chosen.
The first term that differs from A336957 is a(9) = 77. See the examples below.


LINKS

Table of n, a(n) for n=1..70.
Scott R. Shannon, Image of the first 100000 visited numbers on the square spiral. The colors are graduated across the spectrum from red to violet to indicate the relative visit order of the numbers. The starting 1 position is colored white. As the sequence cannot visit primes or prime powers many of the black unvisited squares form the typical diagonalline pattern seen in the Ulam prime spiral.


EXAMPLE

The square spiral used is:
.
1716151413 .
  .
18 543 12 29
    
19 6 12 11 28
   
20 78910 27
 
212223242526
.
a(8) = 33 as a(7) = 12 = 2*2*3 and a(6) = 14 = 2*7, thus a(8) must contain 3 or 2 as a factor but not 2 or 7. As a(6) excludes 2 it must contain 3 as a factor, and another prime other than 3. The closest unvisited number to the starting 1 position that satisfies these conditions is 33.
a(9) = 77 as a(8) = 33 = 3*11 and a(7) = 12 = 2*2*3, thus a(9) must contain 3 or 11 as a factor but not 2 or 3. As a(7) excludes 3 it must contain 11 as a factor, and another prime other than 11. The smallest unvisited number satisfying these conditions is 55, which is sqrt(20) ~ 4.47 units from 1. However 77 is unvisited and also satisfies the conditions, and is only 4 units from 1, thus a(9) = 77. This is the first term that differs from A336957.


CROSSREFS

Cf. A336957, A098550, A335661, A330979, A340783, A064413.
Sequence in context: A287012 A336957 A338055 * A340779 A073838 A337646
Adjacent sequences: A336796 A336797 A336798 * A336800 A336801 A336802


KEYWORD

nonn


AUTHOR

Scott R. Shannon, Jan 27 2021


STATUS

approved



