1,2

First differences give the much older sequence A075568. - njas, Sep 26 2006

Comment from Max Alekseyev, Sep 26 2006: The n-th prime appear in the sequence of first differences (A075568) not later than at the 2n-th position. To see this it is enough to notice that in the original sequence (excluding the first element) odd and even numbers alternate. Therefore from each odd element m the sequence simply jumps to an even element m+p where p is the smallest previously unused prime.

The sequence S is constructed like this:

Start with "1": S = 1, ...

Add the smallest prime not added so far, in order to get a composite:

- Can we add 2? No because 1+2=3 and 3 is prime

- Can we add 3? Yes because 1+3=4 and 4 is composite

So we have now: S = 1,4, ...

Add the smallest prime not added so far in order to get a composite:

- Can we add 2? (smallest available prime); yes because 4+2=6 and 6 is composite

So we have now: S = 1,4,6, ...

- Can we add 5? No because 6+5=11 and 11 is prime

- Can we add 7? No because 6+7=13 and 13 is prime

- Can we add 11? No because 6+11=17 and 17 is prime

- Can we add 13? No because 6+13=19 and 19 is prime

- Can we add 17? No because 6+17=23 and 23 is prime

- Can we add 19? Yes because 6+19=25 and 25 is composite

So we have now: S = 1,4,6,25, ...

- Can we add 5? (smallest available prime); yes because 25+5=30 and 30 is composite

So we have now: S = 1,4,6,25,30, ...

etc.

Sequence in context: A032087 A136591 A009459 this_sequence A028273 A024471 A075277

Adjacent sequences: A123052 A123053 A123054 this_sequence A123056 A123057 A123058

base,easy,nonn

Eric Angelini (eric.angelini(AT)kntv.be), Sep 26 2006

Terms computed by Max Alekseyev.