[seqfan] Sequence

[John Mason]

Hi Seqfans,
Given that almost all integers, in their decimal representation, contain any specific string of digits, I generated: 20327616, 1062880, 2125762, 3188644, 4251526, 5314408, 6377290, 17006110, 18068992, 19131874, as a sequence with offset 0 with the following definition:

a(n) is the lowest positive integer x such that for all y >= x, at least 50% of the positive integers lower than y contain n in their decimal representation.

The sequence is criticisable from at least one point of view: the arbitrary value of 50% in its definition. Should it be proposed all the same?

a(1) is confirmed by the value a(16) in A343839: Semi-one numbers: Positive integers k such that exactly half of the integers 1..k have a 1 in their decimal expansion.

Calculating a(10) has proved beyond my capabilities.

There is a strange coincidence of values between the values I generated and A117862: Number of palindromes (in base 3) below 3^n.

a(1) = A117862(24)
a(2) = A117862(25)
a(3) = A117862(26)
a(6) = A117862(27)
a(9) = A117862(29)

Can anyone explain that?

See also A079004: Least x>=3 such that F(x)==1 (mod 3^n) where F(x) denotes the x-th Fibonacci number (A000045).

john

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