[seqfan] A new sequence
DAN_CYN_J
dan_cyn_j at comcast.net
Fri Jun 14 03:57:25 CEST 2013
Hi all seq. fans.
A summation sieve that never produces an odd prime
but will expose all odd composites.
9+3+2+1+3+2
-2
+4+3+2
-6
+5+4+3+2
-11
+6+5+4+3+2
-17
+7+6+5+4+3+2
-24
+8+7+6+5+4+3+2
-32
+9+8+7+6+5+4+3+2
-41
+10+9+8+7+6+5+4+3+2
etc.
After the 6 first summations a nice pattern appears which
will exclude all primes and exposes all odd composites.
The negations are just the difference of the first negation
of -2 and +4 = next negation of -6
-6 +5 = next negation -11
and so on.
Put into a sequence ---
9,3,2,1,3,2,-2,4,3,2,-6,5,4,3,2,-11,6,5,4,3,2,-17,7,6,5,4,3,2,-24,8,7,6,5,4,3,2,-32,9,8,7,6,5,4,3,2,-41,10,9,8,7,6,5,4,3,2...
This is based on the alternate triangle # matrix.
Where all t(n) - t(n-1,-2-,3,-4...) gives all results
too the right of the triangle number column.
1
3 2
6 3 5
10 4 7 9
15 5 9 12 14
21 6 11 15 18 20
28 7 13 18 22 25 27
36 8 15 21 26 30 33 35
45 9 17 24 30 35 39 42 44
55 10 19 27 34 40 45 49 52 54
...
Eliminate the first 3 columns of this matrix and it produces
the same result as my sequence. Starting in column 4 with (9)
and adding 3 gives 12 in the next row under (9) + 2 gives 14
next to the 12. (14)+1 = 15 in the next row under the 12 and
15+3=18 +2 =20. Then at this point the negations take over for
each succeeding row. 20-2= 18 +4+3+2. -6 +5+4+3+2 etc.
It is indirectly based on the Fermat method of factoring and
because after the first 6 summations and one negation a
pattern develops.
Therefore an algorithm I used in Python was easy to construct.
Slow but effective it just spits out sequential odd primes
not present in an array at the end of the program but along the way
it identifies all the odd composites.
What is interesting also is, no previous odd primes not present
in the array are needed to identify larger odd primes not present
in the array.
Also a few even composites are not captured in the array, like the
even perfect numbers,all 2^n and the 5 Fermat even numbers
that are deficient by two. These are ignored by the algorithm
because only odd blank locations are printed from the array which
produce all the sequential primes at a pre-set ending point.
Cheers,
Dan
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