Angry Naturals

(and Primes)

Hello SeqFans,

Here is another reordering of the Naturals (as I got no reply for my former post -- sigh!);

Imagine that an Angry Natural Number could not stand the presence of his immediate neighbors -- thus asking them to stay away;

The acceptable "stay away" distance would be given by the said angry Natural himself -- this meaning that "5", for instance, would not accept to see "4" and "6" at a shorter distance than 5 integers from him;

Sequence S(1) is one packing of such Angry naturals:

S(1) = 1,4,7,2,10,13,16,3,5,.,.,8,.,.,.,6,11,.,.,.14,9,.,.,.,.,.,.,.,12,.,.,.,.,.,.,15,...

S(1) is build according to the "place the Naturals first" method -- see again the former msg, here.

But we could also use the other building method mentioned on the same page -- "fill the ‘holes’ first" --, which would give:

S(2) = 1,3,5,7,9,2,11,13,4,15,17,6,19,21,8,23,25,27,10,29,31,12, ...

S(2) comes lexicographically first.

Best,

É.

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Franklin T. Adams-Watters replied:

If the requirement is that the neighbor must be at least n away, instead of more than n away, the first packing method gives A072009.

The corresponding sequence for the second approach is not in the database.

A065186 and A065187 are also kind of similar.

One can also look for the lexically first such sequence that is also a self-inverse permutation. If I haven't made any mistakes, for Eric's definition this starts:

1,4,8,2,10,17,25,3,19,5,21,34,13,28, ...

while my variant starts:

1,4,7,2,9,15,3,17,5,19,30,12,25,39,6, ...

For the rule in A065186 (consecutive integers may not be adjacent), this starts:

1,4,6,2,5,3,

and then this pattern repeats with each block 6 larger: a(n+6)=a(n)+6.

None of these are in the database.

Franklin T. Adams-Watters

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Many thanks, Franklin, for the interesting comments (and links).

More comments appeared on SeqFan mailing list a few days later:

If the requirement is that n must be exactly n terms away from n-1, instead of at least n away or more than n away, I found:

1,4,2,131,129,3,5,16,14,12,10,8,6,31,29,27,25,23, ... and the next term a(19) seemed to be difficult.

Surprisingly, the sequence seems to be A057167: “Term in Recaman's sequence A005132 where n appears for first time, or 0 if n never appears”

1, 4, 2, 131, 129, 3, 5, 16, 14, 12, 10, 8, 6, 31, 29, 27, 25, 23, 99734, 7, 9, 11, 13, 15, 17, 64, 62, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 40, 38, 111, 22, 20, 18, 28, 30, 32, 222, 220, 218, 216, 214, 212, 210, 208, 206, 204, 202, 200, 198, 196

So it's easy to see why a(19)=99734 was difficult.

The "angry" property is not mentioned in A057167

John W. Layman

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This breaks for 26, which is the first number that does not occur in A057167; because A005132(26) = A005132(18) = 43 is the first duplication in Recaman's sequence.

Franklin T. Adams-Watters

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Now, what about Angry Primes?

If the Primes were angry -- and only them, not the other Naturals -- we would have no more than the #2 building method: “Fill the next ‘hole’ with the smallest integer not used so far and not leading to a contradiction”.

This, together with the first rule -- “There must be at least P(n) naturals between P(n) and P(n-1)

& there must be at least P(n) naturals between P(n) and P(n+1)” -- would produce:

S(3) = 1,2,4,5,6,8,9,10,11,3,12,14,15,16,17,18,20,21,22,23,7,24,25,26,27,28,30,31,32,33,34,35,13,36,... 19 ... 29 ... 37 ...

Note that, in order for everyone to stay peacefully in S, only a few angry Primes (in yellow) have to jump elsewhere in search of a quite neighborhood...

Best,

É.