[seqfan] Re: S and S complementary : cumulative sum is prime accordingly

[Eric Angelini]

Hello Doug,
thanks for the computation!

I'm convinced this seq is unique (or at least lexico first);
this is how those build themselves automatically (same idea with
the odd/even cumulative sum):

S = 1, 

1 is non-prime -- thus S will be the "non-prime sum self-describing"

I must prolong S with "the smallest available integer not yet
present in S and not leading to a contradiction"; can it be 2?

No:

S = 1, 2, ...

This "2" would say "the sum of the first 2 terms of S is non-prime",
which is false; thus we _transfer "2" to the complementary sequence_:

Scomp = 2,

Now this "2" means "the sum of the first 2 terms of S _is prime_";

So, to extend S, we try "3" which is the "smallest etc." -- but "3"
doesn't fit (1+3 is non-prime) and thus "3" is _immediately_ trans-
ferred to Scomp:

Scomp = 2, 3, ...

Now S can be extended with "4", the "smallest etc." (1+4 is prime):

S = 1, 4, ...
Scomp = 2, 3, ...

Who's instruction have we to obey now? We have correctly executed the
instruction given by "1" and "2"; we consider thus "3" -- and "3" says
that the first 3 terms of S sum up to a prime: can we put "5"?

No:

S = 1, 4, 5, ...
Scomp = 2, 3, ...

We see that the sum 1+4+5 is not prime; we thus _immediately_ transfer
this "5" to Scomp:

S = 1, 4, ...
Scomp = 2, 3, 5, ...

We still have to obey the instruction given by "3"; "6" fits the command
(1+4+6 is prime):

S = 1, 4, 6, ...
Scomp = 2, 3, 5, ...

We must now consider the next instruction, which is given by "4": "The
sum of the first 4 terms of S is non-prime"; does "7" (the "smallest etc.")
fit? Yes, as 1+4+6+7 is non-prime:

S = 1, 4, 6, 7, ...
Scomp = 2, 3, 5, ...

Next instruction is given by "5": "The sum of the first 5 terms of S is
prime". But "8" doesn't fit, nor "9", nor "10"; those three terms are
transferred immediately to Scomp -- and "11" to S:

S = 1, 4, 6, 7, 11, ...
Scomp = 2, 3, 5, 8, 9, 10, ...

etc.

The method:

(1) obey the next instruction 
(2) accept in S the "smallest available integer etc." or transfer it to Scomp
(3) GOTO (1)

 ... gives a unique pair of seq.

I think you have understood this from the beginning! Congrat's and thanks
again!
(I find those kind of pairs fascinating -- am I the only one? Is all
 this too artificial?)

Best,
É.


-----Message d'origine-----
De : seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
De la part de Douglas McNeil
Envoyé : mardi 29 juin 2010 15:32
À : Sequence Fanatics Discussion list
Objet : [seqfan] Re: S and S complementary : cumulative sum is prime accordingly

E. Angelini wrote:

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

I think I can confirm these terms -- it's harder to work backward from
an example than forward from a definition or rule -- but I'm not
convinced the sequence is uniquely specified, unless we select the
lexically first solution.  Don't most of these self-describing
sequences have a "minimal term not leading to a contradiction" rule
attached?

sage: S[:100]
[1, 4, 6, 7, 11, 13, 14, 15, 18, 20, 21, 27, 28, 29, 30, 33, 34, 35,
37, 39, 40, 41, 44, 46, 48, 50, 51, 52, 53, 54, 66, 72, 73, 74, 75,
100, 101, 105, 106, 107, 108, 109, 114, 115, 121, 122, 124, 125, 133,
134, 135, 136, 137, 138, 140, 150, 168, 170, 178, 180, 186, 200, 204,
216, 234, 235, 237, 240, 252, 258, 260, 261, 262, 263, 265, 273, 278,
282, 286, 288, 290, 292, 312, 318, 338, 346, 350, 354, 360, 364, 366,
380, 396, 406, 410, 414, 426, 438, 462, 463]


Doug

-- 
Department of Earth Sciences
University of Hong Kong


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