OFFSET
1,3
COMMENTS
This is the earliest permutation of the nonnegative integers with this property.
EXAMPLE
The spiral begins:
.
16--23--29-5999-33--18
| |
24 5---8--11---7 25
| | | |
39 6 0---1 9 42
| | | | |
19 3---4---2 10 69
| | |
15--13--17--14--12 699
|
... 999--21--26--20
.
The digits of the four integers inside each of the four 2 X 2 squares that contain the initial 0 add up to a prime: 0 + 1 + 2 + 4 = 7, 0 + 4 + 3 + 6 = 13, 0 + 6 + 5 + 8 = 19, 0 + 8 + (1+1) + 1 = 11. This is true for any 2 X 2 square on the (infinite) grid; the upper right corner adds up to the prime 29, for instance: (3+3) + (1+8) + (2+5) + 7 = 29; etc.
All those successive "prime sums" form the hereunder "second-level" spiral:
.
37--19--43 ...
|
43 11--19--19--23
| | |
31 13 7--13 31
| | | |
29 19--11--19 29
| |
29--47--53--29--23
.
Though the terms of this new spiral are not distinct, the sum of the digits inside any 2 X 2 square is prime again; the upper left 2 X 2 square produces the prime 29 = (3+7) + (1+9) + (1+1) + (4+3); the lower left 2 X 2 square produces the prime 43 = (2+9) + (1+9) + (4+7) + (2+9); the lower right 2 X 2 square produces the prime 37 = (1+9) + (2+9) + (2+3) + (2+9); the initial "center square" produces the prime 23 = 7 + (1+3) + (1+9) + (1+1); etc.
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Eric Angelini and Carole Dubois, May 24 2022
STATUS
approved