[seqfan] No 10-sum in any chunk of digits
Eric Angelini
Eric.Angelini at kntv.be
Sun Nov 17 00:13:49 CET 2013
Hello SeqFans,
We want a seq S where it is impossible to
reach exactly 10 by summing any sub-
chain of consecutive digits of S.
S is strictly increasing here:
S = 1,2,3,6,7,8,9,20,21,22,25,26,30,31,40,43,44,45,47,48,49,50,60,61,62,63,65,66,67,68,69,70,71,74,75,76,77,...
We see that S cannot start with:
S = 1,2,3,4,... as 1+2+3+4 = 10
neither with:
S = 1,2,3,5,... as 2+3+5 = 10
Thus S starts with:
S = 1,2,3,6,... as no sub-chain of digits
sums up to 10 yet.
To extend S we always take the
smallest integer bigger than the
previous one and not leading to a
contradiction.
...
T follows the same rule but is
extended with the smallest integer
not yet present in T and not leading
to a contradiction:
T = 1,2,3,6,5,4,7,8,9,20,10,11,12,18,30,13,14,31,34,16,15,17,18,32,21,33,23,35,36,24,25,22,27,26,36,29,38,39,40,...
Is T a permutation of the naturals
without the integers listed in A052224
("Numbers whose sum of digits is 10")?
No -- 11111111, with its 8 digits "1",
cannot be part of T.
...
Why did we consider the 10-sum constraint?
Because if we forbid any chunk of
digits of sum 9, for instance, we must
forbid all 9s in T (well, why not have
a try?)
We could also build respectively an S'
and a T' seq with an 11-sum interdiction (or a 12-sum interdiction -- or a 666-sum ban)...
Best,
É.
Catapulté de mon aPhone
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