A354374 Square spiral on a 2D square lattice, one term per cell, starting at the origin with 0; the digits of the four integers forming any 2 X 2 square add up to a prime and those sums themselves form another infinite 2D square lattice with the same property.
0, 1, 2, 4, 3, 6, 5, 8, 11, 7, 9, 10, 12, 14, 17, 13, 15, 19, 39, 24, 16, 23, 29, 5999, 33, 18, 25, 42, 69, 699, 20, 26, 21, 999, 299, 599, 22, 28, 30, 31, 34, 38, 27, 37, 36, 40, 59, 4999, 43, 32, 35, 41, 49, 102, 47, 69999, 44, 45, 48, 99, 58, 52, 111, 689, 46, 51, 698, 79999, 9999999, 50, 68
Offset: 1
Examples
The spiral begins:
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16--23--29-5999-33--18
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24 5---8--11---7 25
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39 6 0---1 9 42
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19 3---4---2 10 69
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15--13--17--14--12 699
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... 999--21--26--20
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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:
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37--19--43 ...
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43 11--19--19--23
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31 13 7--13 31
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29 19--11--19 29
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29--47--53--29--23
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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.
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