One interesting sequence

 

1, 2, 1, 3, 2, 4, 5, 3, 6, 7, 4, 10, 8, 12, 12, 13, 4, 16, 17, 3, 18, 19, 4, 22, 14, 19, 8, 26, 16, 28, 24, 29, 4, 32, 33, 25, 32, 36, 36, 37, 4, 40, 38, 25, 40, 41, 6, 45, 44, 39, 38, 33, 42, 47, 8, 54, 54, 55, 48, 57, 42, 55, 52, 51, 48, 63, 60, 63, 62, 39…

The sequence’s formula is a(1)=1, a(2)=2, for n>2 a(n)=n-largest prime which divides a(n-1). For “the largest prime which divides n we can use the LPD(n) notation.

For example, let n=6. a(5)=2. Largest prime, which divides 2, is 2. So a(6)=6-2=4.

Is this sequence interesting?

Everyone has the right to his own opinion, and therefore no one is responsible.

I thing that this sequence is very interesting because a(n)=n-LPD(a(n-1)), and no a(n)=n-LPD(n-1). In the latter case, we obtain a similar but less exciting sequence. Because of this peculiarity, I would call the original sequence a “kalach” sequence.

One first observation: if n is a prime then a(n) is often even, but there is at least one exception: a(19) = 17.  Conversely, it appears that if a(n) is a prime, then n is even.

According to my intention, I will report on other interesting things here, so the note may be expanded or modified in the future.

If you have any comments or observations, feel free to let us know.

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