[seqfan] Re: Rank of n in the Odd/Even sequence

[f.firoozbakht at sci.ui.ac.ir]

> I have computed this array yesterday (by hand):
>
>  S(1) =  1, 1,  6,   58,     5829,    58292915, ...
>  S(2) =  2, 1, 11, 1056, 10555528, ...
>  S(3) =  3, 2, 16, 1608, 16080804, ...
>  S(4) =  4, 2, 21, 2111, 21106056, ...
>  S(5) =  5, 3, 27, 2664, 26636332, ...
>  S(6) =  6, 3, 32, 3166, 31661583, ...
>  S(7) =  7, 4, 37, 3719, 37186860, ...
>  S(8) =  8, 4, 42, 4221, 42212111, ...
>  S(9) =  9, 5, 48, 4774, 47742387, ...
> S(10) = 10, 5, 53, 5277, 52767639, ...
>
> Could someone extend this array (to the right) -- or at least S(1)?
>

The Mmca code for writing Si(k) is:

  Si[1]=i;Si[n_]:=Si[n]=(v={};Do[v= Join[v,IntegerDigits[Si[k]]],{k,n-1}];
  Floor[(1+FromDigits[v])/2])

So we can easily find arbitary large terms of each of the sequences Si.

It's interesting that number of digits of S1(k) for k>2 equals to 2^(k-3).
And for i>1 & k >1 number of digits of Si(k) equals to 2^(k-2).

I write 10 terms of S1 and 8 terms of each of the other nine sequences.


S1 = 1, 1, 6, 58, 5829, 58292915, 5829291479146458,
      58292914791464577914645739573229,
      5829291479146457791464573957322929146457395732288957322869786615,
      5829291479146457791464573957322929146457395732288957322869786614
      7914645739573228895732286978661464573228697866144478661434893308,
      5829291479146457791464573957322929146457395732288957322869786614
      7914645739573228895732286978661464573228697866144478661434893307
      7914645739573228895732286978661464573228697866144478661434893307
      3957322869786614447866143489330732286614348933072239330717446654,...

S2 = 2, 1, 11, 1056, 10555528, 1055552805277764,
      10555528052777640527776402638882,
      1055552805277764052777640263888205277764026388820263888201319441,...

S3 = 3, 2, 16, 1608, 16080804, 1608080408040402,
      16080804080404020804040204020201,
      1608080408040402080404020402020108040402040202010402020102010101,...

S4 = 4, 2, 21, 2111, 21106056, 2110605560553028,
      21106055605530281055302780276514,
      2110605560553028105530278027651410553027802765140527651390138257,...

S5 = 5, 3, 27, 2664, 26636332, 2663633213318166,
      26636332133181661331816606659083,
      2663633213318166133181660665908313318166066590830665908303329542,

S6 = 6, 3, 32, 3166, 31661583, 3166158315830792,
      31661583158307916583079157915396,
      3166158315830791658307915791539615830791579153958291539578957698,...

S7 = 7, 4, 37, 3719, 37186860, 3718685968593430,
      37186859685934301859342984296715,
      3718685968593430185934298429671518593429842967150929671492148358,...


S8 = 8, 4, 42, 4221, 42212111, 4221211071106056,
      42212110711060557110605535553028,
      4221211071106055711060553555302821106055355530278555302767776514,...

S9 = 9, 5, 48, 4774, 47742387, 4774238723871194,
      47742387238711937387119361935597,
      4774238723871193738711936193559723871193619355968693559680967799,...

S10 = 10, 5, 53, 5277, 52767639, 5276763876383820,
       52767638763838197638381938191910,
       5276763876383819763838193819191026383819381919098819190969095955,...



  --- Farideh Firoozbakht


Quoting Eric Angelini <Eric.Angelini at kntv.be>:

>
>
> Hello David,
>
>> Such standards can only be enforced by a reasonably objective formal
> editorial review process. In the absence of such a process, the OEIS
> will continue to collect chaff.
>
> ... yes, I fully agree with you.
>
> Is the hereunder idea 'chaff' ?
> ;-)
> Best,
> É.
>
> ---
>
> S = 1, 1, 6, 58, 5829, 58292915, ...
>
> Start with 1
>
> S = 1
>
> Question (a) : is "1" odd or even?
> Answer: "1" is odd
> Question (b) : what is the rank of "1" in the Odd numbers seq.?   
> http://www.research.att.com/~njas/sequences/A005408
> Answer: "1" is the first odd integer, it's rank is = 1
> Action : write this rank as a new term of S :
>
> S = 1, 1,
>
> Question (a) : is "11" (concatenation of all preceding digits) odd or even?
> Answer: "11" is odd
> Question (b) : what is the rank of "11" in the Odd numbers seq.?
> Answer: "11" is the 6th integer, it's rank is = 6
> Action : write this rank as a new term of S :
>
> S = 1, 1, 6,
>
> Question (a) : is "116" (concatenation of all preceding digits) odd or even?
> Answer: "116" is even
> Question (b) : what is the rank of "116" in the Even numbers seq.?   
> http://www.research.att.com/~njas/sequences/A005843
> Answer: "116" is the 58th integer, it's rank is = 58
> Action : write this rank as a new term of S :
>
> S = 1, 1, 6, 58,
>
> Question (a) : is "11658" (concatenation of all preceding digits)   
> odd or even?
> Answer: "11658" is even
> Question (b) : what is the rank of "11658" in the Even numbers seq.?
> Answer: "11658" is the 5829th integer, it's rank is = 5829
> Action : write this rank as a new term of S :
>
> S = 1, 1, 6, 58, 5829, ...
>
> ... etc.
>
> I have computed this array yesterday (by hand):
>
>  S(1) =  1, 1,  6,   58,     5829,    58292915, ...
>  S(2) =  2, 1, 11, 1056, 10555528, ...
>  S(3) =  3, 2, 16, 1608, 16080804, ...
>  S(4) =  4, 2, 21, 2111, 21106056, ...
>  S(5) =  5, 3, 27, 2664, 26636332, ...
>  S(6) =  6, 3, 32, 3166, 31661583, ...
>  S(7) =  7, 4, 37, 3719, 37186860, ...
>  S(8) =  8, 4, 42, 4221, 42212111, ...
>  S(9) =  9, 5, 48, 4774, 47742387, ...
> S(10) = 10, 5, 53, 5277, 52767639, ...
>
> Could someone extend this array (to the right) -- or at least S(1)?
>
> Best,
> É.
>







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