Balenciaga Prime Numbers

At its historic Paris salon, Balenciaga and Demna Gvasalia presented their first Haute Couture collection for Fall 2023 (this was July the 4th, 2023). Here are the prime numbers (the full 59 items are visible there).
The prime numbers in the OEIS
The first 100000 prime numbers












 

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Square my chunks and add

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The idea is to transform the integer a into b and iterate. Cut a into the number of chunks you want, square them, add those squares. Example a = 1023 can be transformed in b  (last column) like this: 1 ² + 0 ² + 2 ² + 3 ² = 1 + 0 + 4 + 9  = 14 = b 1 ² + 0 ² + 23 ²     = 1 + 0 + 529    = 530   10 ² + 2 ² + 3 ²     = 100 + 4 + 9    = 113   10 ² + 23 ²         = 100 + 529      = 629   102 ² + 3 ²         = 10404 + 9      = 10413   1023 ²                              = 1046529   Question #1 Starting with 2 and iterating, what is the shortest path ending in 1? Here are 11 steps (left column) – can you do better? 2    –> 2² = 4 4    –> 4² = 16 16   –> 1² + 6² = 1 + 36 = 37 37   –> 37² = 1369...
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Triples for the new year

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(Dall-e creation ) The hereunder sequence  S  is both a succession of triples of monotonically increasing numbers and a succession of triples of monotonically increasing digits. S  is the lexicographically earliest such sequence with distinct number-triples: S = 0,1,2,0,1,3,0,1,4,0,1,5,0,1,6,0,1,7,0,1,8,0,1,9,0,2,3,0,2,4,0,2,5,0,2,6,0,2,7,0,2,8,0,2,9,0,3,4,0,3,5,0,3,6,0,3,7,0,3,8,0,3,9,0,4,5,0,4,6,0,4,7,0,4,8,0,4,9,0,5,6,0,5,7,0,5,8,0,5,9,0,6,7,0,6,8,0,6,9,0,7,8,0,7,9,0,8,9,1,2,3,1,2,4,1,2,5,1,2,6,1,2,7,1,2,8,1,2,9,1,3,4,1,3,5,1,3,6,1,3,7,1,3,8,1,3,9,1,4,5,1,4,6,1,4,7,1,4,8,1,4,9,1,5,6,1,5,7,1,5,8,1,5,9,1,6,7,1,6,8,1,6,9,1,7,8,1,7,9,1,8,9,2,3,4,2,3,5,2,3,6,2,3,7,2,3,8,2,3,9,2,4,5,2,4,6,2,4,7,2,4,8,2,4,9,2,5,6,2,5,7,2,5,8,2,5,9,2,6,7,2,6,8,2,6,9,2,7,8,2,7,9,2,8,9,3,4,5,3,4,6,3,4,7,3,4,8,3,4,9,3,5,6,3,5,7,3,5,8,3,5,9,3,6,7,3,6,8,3,6,9,3,7,8,3,7,9,3,8,9,4,5,6,4,5,7,4,5,8,4,5,9,4,6,7,4,6,8,4,6,9,4,7,8,4,7,9,4,8,9,5,6,7,5,6,8,5,6,9,5,7,8,5,7,9,5,8,9,6,7,8,6,7,9,6,8,9,7,...
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Du recuit simulé

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Toujours trouvé que  cette expression  était magnifique ! Et voilà qu'elle devient (un peu) mienne grâce au triangle ci-dessous (grâce aussi à Jean-Charles M . et  Mark K .)  Tout a commencé par un message (raté) sur Math-Fun et sa correction : Hello Math-Fun, no cell of the hereunder triangle shares any of its digits with another cell of the same rank or column: +-----+-----+-----+-----+-----+-----+-----+-----+-----+ |  10 |  20 |  30 |  40 |  50 |  60 |  70 |  80 |  90 |        +-----+-----+-----+-----+-----+-----+-----+-----+-----+ |  22 |  11 | 777 | 666 | 888 | 333 | 444 |   9 |                +-----+-----+-----+-----+-----+-----+-----+-----+ |  33 |   8 | 555 | 111 | 222 |  47 |  69 |                +-----+-----+-----+-----+-----+-----+-----+ |  44 |  ...
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