[seqfan] Sums, chunks and divisions -- twice
Eric Angelini
Eric.Angelini at kntv.be
Fri Nov 28 22:56:01 CET 2014
Hello SeqFans,
please set the width of your "reading window" to 90 characters without automatic "carriage return"-- else the explanations might be quite difficult to follow...
(or copy/paste the whole damn mail into Word -- then push the margins --equivalent and better pgms exists ).
Here is C:
C=1,2,4,3,5,6,7,8,9,71,10,11,12,13,14,15,16,17,19,18,20,21,41,22,23,24,25,33,26,...
This will ring a bell now, but wait:
a) Insert a ":" immediately after the 1st integer, then (from there) another ":" after 2 integers, then a ":" after 4 integers, then after 3 integers, etc. (the marks are inserted according to the terms of the sequence C itself); we get:
C=1:2,4:3,5,6,7:8,9,71:10,11,12,13,14:15,16,17,19,18,20:21,41,22,23,24,25,33:26,...
We see now a succession of :chunks: -- where the sum of the integers in each chunk is divisible by the first integer of the said chunk (let’s check with S=sum; D=division; Q=quotient; Z being the size of the chunk, given by C itself):
Z=1 2 4 3 5 6 7 ..8..
C=1:2,4:3,5,6,7:8,9,71:10,11,12,13,14:15,16,17,19,18,20:21,41,22,23,24,25,33:26,...
S=1 +6 +21 +88 +60 +105 +182
D=1 6/2 21/3 88/8 60/10 105/15 182/21
Q=1 =3 =7 =11 =6 =7 =8
This is not new (see my former post). But wait... Let’s start with C again:
C=1,2,4,3,5,6,7,8,9,71,10,11,12,13,14,15,16,17,19,18,20,21,41,22,23,24,25,33,26,...
b) Insert now a "*" immediately after the 1st _digit_, then (from there) another "*" after 2 _digits_, then "*" after 4 _digits_, then after 3 digits, etc. -- the stars are inserted according to the terms –not the digits!– of the sequence C itself. We get:
C=1*2,4*3,5,6,7*8,9,7*1,10,11*12,13,14*15,16,17,1*9,18,20,21,4*1,22,23,24,25,33*26,...
Miracle! We see again a succession of *chunks* -- where the sum of the _digits_ in each chunk is divisible by the first _digit_ of the said chunk (let’s check with S=sum; D=division; Q=quotient; Z being the size of the digit-chunk, given by C itself):
Z=1 2 4 3 5 6 7 8 9 ..71..
C=1*2,4*3,5,6,7*8,9,7*1,10,11*12,13,14*15,16,17,1*9,18,20,21,4*1,22,23,24,25*33,26,...
S=1 +6 +21 +24 +4 +12 +22 +27 +175
D=1 6/2 21/3 24/8 4/1 12/1 22/1 27/9 175/1
Q=1 =3 =7 =3 =4 =12 =22 =3 =175
C was always extended with the smallest integer not yet present in C, respecting both (a) and (b) -- and not forcing a _digit-chunk_ to start with zero (else a division by zero would arise)
Is C a permutation of the integers >0? I guess yes.
Best,
É.
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