A336946 a(1) = 1; for n > 1, a(n) is the next square spiral number not already used such that a(n) shares a factor with a(n-1) and also with the adjacent number on the inner spiral arm if such a number exists.

1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 20, 14, 7, 28, 16, 18, 15, 21, 24, 22, 11, 33, 30, 25, 35, 40, 45, 36, 26, 42, 27, 63, 48, 32, 34, 50, 38, 54, 39, 51, 60, 44, 46, 66, 55, 65, 70, 49, 56, 52, 58, 72, 57, 76, 62, 78, 13, 117, 69, 75, 80, 64, 68, 74, 37, 148

Offset: 1

Author

Scott R. Shannon, Aug 08 2020

Keywords

• nonn

Comments

This is a variation of the EKG sequence A064413 where the numbers are written on the square spiral such that each new number must share a common factor with not only the previous number but also with the adjacent inner spiral number, in one of the four axial directions, if such a number exists. This additional restriction causes the numbers to violate some of the patterns the numbers form in the standard EKG sequence, e.g., an odd prime p does not need to be preceded by 2p or followed by 3p, and the primes do not appear in increasing order.

For the first 100000 terms the smallest unseen number is 433.

Examples

			The spiral begins
                                .
                                .
   38--50--34--32--48--63--27  78
    |                       |   |
   54  15--18--16--28---7  42  62
    |   |               |   |   |
   39  21   3---6---4  14  26  76
    |   |   |       |   |   |   |
   51  24   9   1---2  20  36  57
    |   |   |           |   |   |
   60  22  12---8--10---5  45  72
    |   |                   |   |
   44  11--33--30--25--35--40  58
    |                           |
   46--66--55--65--70--49--56--52
.
a(1)-a(8) = 1,2,4,6,3,9,12,8. The adjacent inner spiral number is 1 which all numbers share a factor with so the numbers are the same as A064413(n).
a(9) = 10. This is the first number that must have a common factor with two numbers, the previous number a(8) = 8 and the adjacent spiral number a(2) = 2. The lowest unused number satisfying this requirement is 10.
a(10) = 5. As this number is on the corner of a square spiral arm it only needs to share a divisor with a(9) = 10. The lowest unseen number satisfying this is 5.
a(11) = 20.  This number must have a common factor with the previous number a(10) = 5 and the adjacent spiral number a(2) = 2. The lowest unused number satisfying this requirement is 20. This is also the first number to differ from  A064413 which only needs to find the lowest unused number sharing a factor with 5, which is 15.
		

Links

• Scott R. Shannon, Line graph of the first 1000 terms.

Crossrefs

Cf. A064413, A073734, A253279, A257112.