Self-Parity/Antiparity Sequences & Numbers
Hello
SeqFans,
321
is a “self-parity number”.
Its first digit, 3, says: - the 3rd digit has my parity (which is
true, “1” is odd as “3” is)
Its second
digit, 2, says: - the 2nd digit has my parity (which is true, “2”
and “2” are even)
Its third digit,
1, says: - the 1st digit has my parity (which is true, “3” is odd as
“1” is)
A
“self-parity number” is a number whose digit ‘speak’ and assert true statements
about their parity.
The
sequence of such self-parity numbers seems to start like this:
S = 1, 11, 12, 22, 111, 113,
121, 122, 123, 131, 133, 222, 311, 313, 321, 323, 331, 333, 1111, 1113, 1131, ...
(this is not A105945)
Could
someone please check and compute a hundred terms more or so?
__________
2343
is a “self-antiparity number”. Its first digit, 2, says: - the
2nd digit has not my parity
(which is true, “3” is odd and “2” is even)
Its second
digit, 3, says: - the 3rd digit has not my parity
(which is true, “4” is even and “3” is odd)
Its third
digit, 4, says: - the 4th digit has not my parity
(which is true, “3” is odd and “4” is even)
Its last digit, 3, says: - the 3rd
digit has not my parity
(which is true, “4” is even and “3” is odd)
A
“self-antiparity number” is a number whose digit
‘speak’ and assert true statements about their non-parity.
The
sequence of such self-antiparity numbers seems to
start like this:
S = 21, 211, 212, 232, 332,
2111, 2112, 2121, 2123, 2141, 2143, 2321, 2322, 2341, 2343,
...
(this is not in
the OEIS)
Could
someone please check and compute a hundred terms more or so?
---
Alois Heinz was fast in replying:
Hello
Éric,
Self-antiparity numbers:
S = 21, 211, 212, 232, 332, 2111, 2112, 2121,
2122, 2123, 2141, 2143, 2321, 2322, 2323, 2341, 2343, 3322, 3323, 3343, 3443, 4111,
4121, 4123, 4141, 4143, 4321, 4323, 4341, 4343, 4411, 4441, 4443, 21111, 21112,
21114, 21121, 21122, 21152, 21154, 21211, 21212, 21213, 21214, 21221, 21222,
21223, 21231, 21232, 21233, 21234, 21252, 21254, 21411, 21412, 21413, 21414,
21431, 21432, 21433, 21434, 21452, 21454, 21512, 21514, 21522, 21552, 21554,
23211, 23212, 23213, 23214, 23221, 23222, 23223, 23231, 23232, 23233, 23234, 23252,
23254, 23411, 23412, 23413, 23414, 23431, 23432, 23433, 23434, 23452, 23454,
25112, 25114, 25122, 25152, 25154, 25212, 25214, 25222, 25232, 25234, 25252,
25254, 25412, 25414, 25432, 25434, 25452, 25454, 25512, 25514, 25522, 25552,
25554, 33222, 33223, 33232, 33233, 33234, 33252, 33254, 33432, 33433, 33434,
33452, 33454, 34433, 34434, 34454, 35222, 35232, 35234, 35252, 35254, 35432, 35434,
35452, 35454, 41111, 41112, 41114, 41152, 41154, 41211, 41212, 41213, 41214,
41231, 41232, 41233, 41234, 41252, 41254, 41411, 41412, 41413, 41414, 41431,
41432, 41433, 41434, 41452, 41454, 41512, 41514, 41552, 41554, 43211, 43212,
43213, 43214, 43231, 43232, 43233, 43234, 43252, 43254, 43411, 43412, 43413,
43414, 43431, 43432, 43433, 43434, 43452, 43454, 44111, 44114, 44154, 44411, 44413,
44414, 44431, 44433, 44434, 44454, 44514, 44554, 45112, ...
Alois
Many
thanks, Alois!
__________
Let’s
try now to build, in the same spirit, a (monotonically increasing) “self-parity sequence”.
In
this sequence, the a(n)th
term says that the a(n)th digit of the
sequence has the same parity as a(n):
S = 1, 2, 3, 4, 5, 6, 7, 8,
9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46,
48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86,
88, 91, 93, 95, 97, 99, 102, 103, 107, ...
(this is not A059708)
Could
someone please check and compute a thousand terms more or so?
__________
In a
similar way, we can try to build a (monotonically increasing) “self-antiparity
sequence”.
In this
sequence, the a(n)th
term says that the a(n)th digit of the
sequence has not the same parity as a(n):
S =
2,3,4,5,6,7,8,9,20,21,22,23,24,25,27,29,40,41,42,43,44,45,46,47,49,61,63,64,65,66,67,68,69,70,...
(building method: “always take the smallest available integer
–bigger than the previous one– not leading to a contradiction”. This sequence
is not easy to calculate by hand!)
If
we drop the “monotonically increasing” constraint, we might have:
S = 2,1,4,3,6,5,8,7,11,20,10,17,21,23,24,25,27,29,40,16,18,19,26,37,...
(building technique: “always take the smallest available
integer not yet present in S and not
leading to a contradiction”. This sequence is almost impossible to calculate by
hand!)
Again,
would someone please check and compute more terms for both sequences?
Best,
É.