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a(n) = number of decimal digits of pi, Pi, starting after the decimal point, that give an average value exactly equal to a whole number.
Partial sum of a(n) digits of decimal expansion of pi equals an integer N x * a(n).
If pi Pi is normal then average digit in limit = 4.5.
Related to drunkard's walk? How many times does the drunkards drunkard's walk cross the x-axis?
No more terms below 5,000,000 [From _. - _Harvey P. Dale_, Apr 07 2010]
a(2)=3 because the first 3 decimal places of pi, Pi, the digits are 1+4+1, has an integer average of 6/3 = 2.
1 = 1x1, 1*1, compressed .. . 11
6 = 2x3, 2*3, compressed ... 23
20 =4x5, 4*5, compressed ... 45
28 =4x7, 4*7, compressed ... 47
36 =4x9, 4*9, compressed ... 49
65 =5x13, 5*13, compressed ... 513
100 = 5x20, 5*20, compressed ... 520.
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] [From (* 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 ] ] [ From _(* _Harvey P. Dale_, Apr 07 2010 ]*)
_Donald S. McDonald (paragraph(AT)mensa.org.nz), _, Mar 22 2007
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editing
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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 = 1x1, compressed .. 11
6 = 2x3, compressed ... 23
20=4x5, compressed ... 45
28=4x7, compressed ... 47
36=4x9, compressed ... 49
65=5x13, compressed ... 513
100 = 5x20, compressed ... 520.
Digit sum (after decimal point) = N x a(n).
0+9=1+8=2+7=3+6=4+5=9.
No more terms below 30000 [From J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010]
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 = 1x1, compressed .. 11
6 = 2x3, compressed ... 23
20=4x5, compressed ... 45
28=4x7, compressed ... 47
36=4x9, compressed ... 49
65=5x13, compressed ... 513
100 = 5x20, compressed ... 520.
base,fini,more,nonn
Corrected and extended - by J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
Edited by Max Alekseyev, Oct 14 2012
approved
editing
No more terms below 5,000,000 [From _Harvey P. Dale (hpd1(AT)nyu.edu), _, Apr 07 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 ] ] [ From _Harvey P. Dale (hpd1(AT)nyu.edu), _, Apr 07 2010 ]
1, 3, 5, 7, 9, 13, 18, 20, 62
No more terms below 30000 [From J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010]
No more terms below 5,000,000 [From Harvey P. Dale (hpd1(AT)nyu.edu), Apr 07 2010]
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] [From 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 ] ] [ From Harvey P. Dale (hpd1(AT)nyu.edu), Apr 07 2010 ]
base,fini,more,nonn,new
Corrected and extended - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
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, 20
1,2
Partial sum of a(n) digits of decimal expansion of pi equals an integer N x a(n).
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 = 1x1, compressed .. 11
6 = 2x3, compressed ... 23
20=4x5, compressed ... 45
28=4x7, compressed ... 47
36=4x9, compressed ... 49
65=5x13, compressed ... 513
100 = 5x20, compressed ... 520.
Digit sum (after decimal point) = N x a(n).
If pi is normal then average digit in limit = 4.5.
0+9=1+8=2+7=3+6=4+5=9.
Related to drunkard's walk? How many times does the drunkards walk cross the x-axis?
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.
base,fini,more,nonn
Donald S. McDonald (paragraph(AT)mensa.org.nz), Mar 22 2007
approved