A354453 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 2 X 2 square of numbers sums to a prime, and that prime is unique for all such squares. Start with a(1) = 0.

0, 1, 2, 4, 3, 6, 5, 8, 14, 7, 9, 17, 10, 12, 19, 21, 11, 18, 16, 32, 13, 23, 25, 20, 30, 15, 27, 40, 31, 43, 22, 28, 39, 37, 36, 41, 24, 51, 57, 48, 35, 69, 26, 49, 66, 53, 65, 58, 76, 29, 61, 88, 38, 90, 33, 113, 34, 54, 123, 67, 86, 74, 100, 98, 42, 75, 91, 70, 96, 102, 71, 117, 44, 106, 126

Offset

1,3

Comments

This is a variation of A337116 where the same rules apply except that the primes generated by all 2 X 2 square sums must be unique. This leads to the terms having a far greater variation in value while being concentrated along a central line which shows wave-like variations in density. See the linked image. The reason for this behavior is unknown.

See A354460 for the successive prime sums formed by each completed 2 X 2 square of numbers.

Links

Table of n, a(n) for n=1..75.

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.

Example

The spiral begins
                                .
                                .
   24--41--36--37--39--28--22 113
    |                       |   |
   51  11--21--19--12--10  43  33
    |   |               |   |   |
   57  18   3---4---2  17  31  90
    |   |   |       |   |   |   |
   48  16   6   0---1   9  40  38
    |   |   |           |   |   |
   35  32   5---8--14---7  27  88
    |   |                   |   |
   69  13--23--25--20--30--15  61
    |                           |
   26--49--66--53--65--58--76--29
.
.
a(9) = 14 as this completes a 2 X 2 square of numbers 0,1,8,14 which sum to 23, a prime, and 14 is the smallest unused number to form a prime sum that has not occurred before. Note that 10 is unused and would form a prime sum of 19, see A337116, but 19 was formed previously by the square 6,0,5,8, so cannot be used. This is the first term to differ from A337116.

Crossrefs

Cf. A354460, A337116, A354441, A257339, A354434, A000040.

Sequence in context: A227114 A354373 A354374 * A281120 A246682 A143691

Adjacent sequences: A354450 A354451 A354452 * A354454 A354455 A354456

Keyword

nonn,look

Author

Scott R. Shannon, May 30 2022

Status

approved