A279035 Left-concatenate zeros to 2^(n-1) such that it has n digits. In the regular array formed by listing the found powers, a(n) is the sum of (nonzero) digits in column n.

1, 2, 4, 9, 9, 9, 8, 19, 9, 8, 17, 27, 27, 27, 28, 17, 26, 35, 45, 45, 46, 37, 25, 44, 53, 65, 42, 72, 74, 52, 70, 90, 92, 74, 53, 62, 72, 70, 93, 61, 81, 80, 89, 100, 91, 80, 91, 79, 99, 99, 99, 98, 107, 117, 118, 106, 130, 86, 123, 155, 137, 117, 118, 105, 136

Offset: 1

Author

David A. Corneth, Dec 03 2016

Keywords

• nonn
• base

Comments

After carries, this is the decimal expansion of Sum_{i>=0} 0.2^i = 1.25. For n > 2, the 10^0's digit of a(n) + the 10^1's digit of a(n+1) + ... + the 10^m's digit of a(n+m) = 9 for some finite m.

Conjecture: a(n) ~ c*n where c ~= 1.93.

Conjecture: lim_{n->infinity} a(n)/n = (9/2)*log_5(2) =

1.93804... - Jon E. Schoenfield, Dec 09 2016

Examples

			1
.2
. 4
. .8
. .16
. . 32
. . .64
. . .128
. . . 256
. . . .512
. . . .1024
The sum of digits of the first column is 1. Therefore, a(1) = 1.
The sum of digits in column 4 is 8 + 1 = 9. Therefore, a(4) = 9.
With the powers of 2 listed above, we can find n up to n = 7. For n > 8, some digits from 2^m compose a(n) for m > 10.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A016134.

Programs

Mathematica

f[n_, b_] := Block[{k = n}, While[k < n + Floor[ k*Log10[b]], k++]; Plus @@ Mod[ Quotient[ Table[ b^j*10^(k - j), {j, n -1, k}], 10^(k - n +1)], 10]]; Table[f[n, 2], {n, 65}]
 (* Robert G. Wilson v, Dec 03 2016 *)