[seqfan] Creating another sequence spawned by A007310
DAN_CYN_J
dan_cyn_j at comcast.net
Sat Jul 13 04:56:58 CEST 2013
See A007310 as a reference.
This is the same sequence but with a negative value every other term
which brings out another sum sequence with interesting consequences.
This sequence A007310 is a (monic polynomial) of degree 2
where each succeeding term that is NOT prime
is just an ABS multiple of previous primes or composites in the sequence.
The sequence starts with prime (5)
5,-7,11,-13,17,-19,23,-25,29,-31,35,-37,41,-43,47,-49,53,-55,59,-61,65,-67,71,-73,77,-79,83,-85,89,-91,95,-97,101,-103,107,-109,113,-115,119,-121,125,-127...
The sum of the right diagonal of each corresponding term
from above are eventually scattered in the above sequence as the +members only
which are 2(mod 3) only.
The sums of the right diagonal in the monic polynomial
at each point = sum seq. =
5,5,35,17,53,23,71,29,89,35,107,41,125,47,143,53,161,59,179,65,197,71,215,77,233,83,251,89,269,95,287,101,305,107,323,113,341,119,359,125,377...
Not in OEIS but directly related to A007310.
As noted above in this right diagonal sum seq., when a lager integer
is flanked by two smaller integers later in the sequence this same
integer will be flanked by two larger integers.
Where the sum of the right diagonal that is the smallest term
(The sum that sits between two larger sum terms)
will be the next term in the sequence of the corresponding first sequence A007310.
Eg: 9th term = 35 in sum seq. and 35 is 10th term in A007310.
The sum sequence's terms are all sum(S) S==2(mod 3).
So all S==1(mod 3) primes and composites will not be listed in the sum sequence
but both n==1or2(mod 3) primes and composites will be listed in A007310.
This sequence A007310 will include all the primes>3 and
all composites with multiple prime factors where these factors are >3 ---->oo
These also include all primes (p) >3 where p^n and n=1,2,3,4,5...
Are there other things I am not observing about this sum sequence?
Dan
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