A106043 First digit other than 9 in the fractional part of the decimal expansion of (1/1000^n)^(1/1000^n).

0, 3, 8, 7, 7, 6, 5, 5, 4, 3, 3, 2, 1, 1, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset

0,2

Comments

From Jon E. Schoenfield, Feb 21 2021: (Start)

Even if each term of this sequence were incremented by 1 (to give them a minimum of 1 and a maximum of 9), their distribution would not follow Benford's law (nor does the related sequence whose n-th term is the first nonzero digit in the fractional part of the decimal expansion of 1 - (1/1000^n)^(1/1000^n)).

For n > 1, a(n) - a(n-1) = 0 or -1 except when a(n) = 8 and a(n-1) != 8; this situation occurs at n = 2, 15, 145, 1448, ..., and from each such value of n until the next, all non-9 digits occur with very nearly equal frequency. E.g., the digits 0..8 occur with frequencies

[ 1, 0, 0, 1, 0, 0, 0, 0, 0] in a(0)..a(1),

[ 1, 2, 1, 2, 1, 2, 1, 2, 1] in a(2)..a(14),

[ 14, 15, 14, 15, 14, 15, 14, 15, 14] in a(15)..a(144),

[145,144,145,145,145,144,145,145,145] in a(145)..a(1447).

(End)

Links

Table of n, a(n) for n=0..86.

Index entries for sequences related to Benford's law

Example

From Jon E. Schoenfield, Feb 21 2021: (Start)
In each of the decimal expansions in the table below, each 9 has been replaced by an underscore character (to make it easy to see at a glance the first non-9 digit in the fractional part):
.
         decimal expansion of (1/1000^n)^(1/1000^n)
   n             with each 9 replaced by "_"             a(n)
  --  -------------------------------------------------  ----
   0  1.00000000000000000000000000000000000000000000...    0
   1  0.__3116048420_337715764260768851547466351_162...    3
   2  0.____8618458487576222544_06332_28167145404344...    8
   3  0._______7_276734377780460834_3251023824_06354...    7
   4  0.__________72368_78884453188455735031275_4061...    7
   5  0._____________6546122360508__11203267556__264...    6
   6  0.________________5855346832610717854658364771...    5
   7  0.___________________516457130471250406367_124...    5
   8  0.______________________44737_57768142_0358356...    4
   9  0._________________________3783020248_16076653...    3
  10  0.____________________________30_2244721017862...    3
  11  0._______________________________240146_1_311_...    2
  12  0.__________________________________17106_3665...    1
  13  0._____________________________________101__18...    1
  14  0.________________________________________032_...    0
  15  0.__________________________________________8_...    8
(End)

Prog

(PARI) zerotozero(n) = { local(x, y, z, v, j); for(x=0, n, y=1000^x; v=(1./y)^(1/y); z=Vec(Str(v)); for(j=3, n, if(z[j]<>"9", print1(z[j]", "); break) ) ) }

Crossrefs

Sequence in context: A276120 A197842 A153020 * A294833 A202478 A193720

Adjacent sequences: A106040 A106041 A106042 * A106044 A106045 A106046

Keyword

base,nonn

Author

Cino Hilliard, May 06 2005

Status

approved