[seqfan] Progression of initial digits of powers

Allan Wechsler acwacw at gmail.com
Mon Nov 24 20:08:28 CET 2014


A008952 shows the first digit of 2^n.  All 9 nonzero digits occur, of
course. (7 first appears at the front of 2^46, and 9 follows shortly in
2^53.)

Suppose we consider length-2 subsequences of A008952. These start (1,2),
(2,4), (4,8) ... A little arithmetic shows that 13 of these pairs are
possible. As a consequence of the irrationality of log[10]2, all 13 do
appear in the sequence.

If we proceed to length-3 subsequences, there are 17 possible triplets of
consecutive initial digits, and they all occur.

21 length-4 subsequences occur, and if I didn't make any mistakes in my
hand calculations, 25 length-5 subsequences.

These statistics would be the same for any geometric sequence with constant
ratio 2; the powers of two are just an easy, familiar example.

The sequence 9, 13, 17, 21, 25 occurs a couple of dozen times in OEIS, but
never with this interpretation. I have a vague suspicion that it is in fact
the tail of A016813 or (equivalently) A004766, but I have no proof.

The same trick could produce initial-digit-subequence counts for any base
above 2 and for any vaguely-exponential source sequence.


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