1,3

This is the earliest permutation of the nonnegative integers with this property.

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.

Cf. A337115, A337116, A337117, A337368, A354372, A354373, A354375.

Sequence in context: A281117 A227114 A354373 * A354453 A281120 A246682

Adjacent sequences: A354371 A354372 A354373 * A354375 A354376 A354377

base,nonn

Eric Angelini and Carole Dubois, May 24 2022

approved