Sequence with equal digit sums

For some nn and kk let a1,a2,...,ana_{1}, a_{2}, ..., a_{n} be a sequence where:

  • aia_{i} is a positive integer

  • a1=ka_{1} = k

  • ai+1=2×aia_{i+1} = 2 \times a_{i}

  • Digit sums of all aia_{i } are equal (base 1010)

Can we construct such sequence for any positive integer nn (kk is of your choice)?

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