[seqfan] summation\negation sieve sequence

[Daniel Joyce]

Hi all,

The summation\negation sequence that eliminates all primes and powers of 2.
Also all even perfect numbers and even integers that are deficient by (2).
The 5 deficient by (2) even numbers that are related to the 5 Fermat
primes are --

3,10,136,32896,2147516416 which is A191363
These are the only known deficient by 2 integers.

The first 8 integers that are eliminated from this sequence because --

1 is not an integer.
2 prime
3 prime and a deficient by 2 integer.
4 2^2
5 prime
6 even perfect number.
7 prime
8 2^3

So this summation sequence begins --

9,3,2,3,2,1,-2,4,3,2,-6,5,4,3,2,-11,7,6,5,4,3,2,-18,8,7,6,5,4,3,2,-26,9,8,7,6,5,4,3,2,
-35,10,9,8,7,6,5,4,3,2,-45...

A pattern emerges where the fist negation (-2) = 18
The next negation (-6) = 21 the next (-11) = 24 and so-on.
After the first negation (-2) the other summation pattern following each
negation
is apparent.
t(4)-1 = sum 9
t(5)-1 = sum 14
t(6)-1 = sum 20
t(7)-1 = sum 27
...
I devised a sieve program to run after a predetermined number of iterations
of the
above sequence to just list the odd empty locations and only the sequential
primes listed.

Dan