A126278 a(n) = number of decimal digits of Pi, starting after the decimal point, that give an average value exactly equal to a whole number.

1, 3, 5, 7, 9, 13, 18, 20, 62

Offset

1,2

Comments

Partial sum of a(n) digits of decimal expansion of pi equals an integer N * a(n).

If Pi is normal then average digit in limit = 4.5.

Related to drunkard's walk? How many times does the drunkard's walk cross the x-axis?

No more terms below 5,000,000. - Harvey P. Dale, Apr 07 2010

Links

Table of n, a(n) for n=1..9.

Example

a(2)=3 because the first 3 decimal places of Pi, the digits are 1+4+1, has an integer average of 6/3 = 2.
Pi = 3.14159 26...
Digit sums 1, 5=1+4, 6=1+4+1, 11, 20, 22, 28...
Number of digits =1, 2, 3, 4, 5, 6, 7.
Average 1, 2.5, 2, 2.75, 4, 3.7,4...
Average is a whole number: 1, 2, 4, 4 ...
When number of digits equals a(n) = 1 3 5 7 9 13 20.
     1 = 1*1,  compressed ... 11
     6 = 2*3,  compressed ... 23
    20 = 4*5,  compressed ... 45
    28 = 4*7,  compressed ... 47
    36 = 4*9,  compressed ... 49
    65 = 5*13, compressed ... 513
   100 = 5*20, compressed ... 520.

Mathematica

Block[{i = 30000, z = RealDigits[Pi - 3, 10, 30000][[1]], lst = {}}, While[z != {}, If[Divisible[Total[z], i], PrependTo[lst, i]]; i--; z = Most@z; ]; lst] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
lst=Accumulate[ Rest[ RealDigits[ N[ \[ Pi ], 5000001 ] ][ [ 1 ] ] ] ]; Transpose[ Select[ Partition[ Flatten[ Table[ {n, (Take[ lst, {n} ])/n}, {n, 5000000} ], 2 ], 2 ], IntegerQ[ #[ [ 2 ] ] ]& ] ][ [ 1 ] ] (* Harvey P. Dale, Apr 07 2010 *)

Crossrefs

Sequence in context: A200975 A370452 A058871 * A121259 A211139 A089228

Adjacent sequences: A126275 A126276 A126277 * A126279 A126280 A126281

Keyword

base,more,nonn

Author

Donald S. McDonald, Mar 22 2007

Extensions

Corrected and extended by J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

Edited by Max Alekseyev, Oct 14 2012

Status

approved