[seqfan] a(a(n)+a(n+1)) has property X

[Eric Angelini]

Hello SeqFans,
S is the lexico-first permutation of the positive integers with the property « a(a(n)+a(n+1)) is even »

S = 1, 2, 4, 3, 5, 6, 8, 10, 7, 9, 12, 11, 13, 14, 15, 16, 18, 20, 17, 19, 22, 21, 24, 26, 23, 25, 28, 27, 30, 29, 32, 31, 33, 34, 35, 36, 38, 40, ...

In other words:
a)  take two adjacent integers x and y in S
b)  let (x + y) = z
c)  a(z) is even.
S was extended with the smallest integer not yet in S and not leading to a contradiction.

Testing the formula:
n = 1  2  3  4  5  6  7   8  9  10 11  12  13  14  15  16  17  18  19  20  21  22 ...
S = 1, 2, 4, 3, 5, 6, 8, 10, 7, 9, 12, 11, 13, 14, 15, 16, 18, 20, 17, 19, 22, 21,...

for n=1 then a(1)=1 and a(2)=2 and a(sum) reads a(1+2) reads a(3) which is 4 (even);
for n=2 then a(2)=2 and a(3)=4 and a(sum) reads a(2+4) reads a(6) which is 6 (even);
for n=3 then a(3)=4 and a(4)=3 and a(sum) reads a(4+3) reads a(7) which is 8 (even);
for n=4 then a(4)=3 and a(5)=5 and a(sum) reads a(3+5) reads a(8) which is 10 (even);
... etc.
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Remark #1:
The seq T, where a(a(n)+a(n+1)) is always odd is already in the OEIS:
https://oeis.org/A000027
;-D
But if we force a(1)=2 we then get again a permutation of A000027:

T' = 2, 1, 3, 5, 4, 6, 7, 9, 11, 13, 8, 10, 15, 12, ,14, 17, 16, 19, 18, 21, 23, 20, 22, 25, 27, 29, 31, 24, 26, 28, 33, ...

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Remark #2:
The seq P, where a(a(n)+a(n+1)) is always prime is also a permutation of of A000027:

P = 1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 13, 10, 17, 12, 19, 14, 15, 23, 29, 31, 16, 37, 41, 18, ...

Best,
É.