packing density of jerblets question

That sequence 1,2,4,6,9,12,16,20,25,30,36,42,49,56,64, above is in oeis:

Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4)

primes mentioned regarding this sequeuce on that page: " Alternative statement of Oppermann's conjecture: For n>2, there is at least one prime between a(n) and a(n+1). - Ivan N. Ianakiev, May 23

2013. [This conjecture was mentioned in A220492, A222030. - Omar E. Pol, Oct 25 2013]

For any given prime number, p, there are an infinite number of a(n) divisible by p, with those a(n) occurring in evenly spaced clusters of three as a(n), a(n+1), a(n+2) for a given p. The divisibility of all a(n) by p and the result are given by the following equations, where m >= 1 is the cluster number for that p: a(2m*p)/p = p*m^2 - m; a(2m*p +

1)/p = p*m^2; a(2m*p + 2)/p = p*m^2 + m. The number of a(n) instances between clusters is 2*p - 3. - Richard R. Forberg, Jun 09 2013 "

or also it is this sequence too: Numbers n such that ceil(sqrt(n)) divides n

cheers, Jamie

I did a similar test of finding jerblets in prime number sequences and came across another OEIS sequence:

2,3 has 0 jerblets.. 2,3,5 has 0 jerblets.. (+0) 2,3,5,7 has 1 jerblets.. (+1) 2,3,5,7,11 has 3 jerblets.. (+2) 2,3,5,7,11,13 has 5 jerblets.. (+2) 2,3,5,7,11,13,17 has 9 jerblets.. (+4) 2,3,5,7,11,13,17,19 has 11 jerblets.. (+2) 2,3,5,7,11,13,17,19,23 has 14 jerblets.. (+3) 2,3,5,7,11,13,17,19,23,29 has 18 jerblets.. (+4) 2,3,5,7,11,13,17,19,23,29,31 has 20 jerblets.. (+2) 2,3,5,7,11,13,17,19,23,29,31,37 has 24 jerblets.. (+4) 2,3,5,7,11,13,17,19,23,29,31,37,41 has 29 jerblets.. (+5) 2,3,5,7,11,13,17,19,23,29,31,37,41,43 has 33 jerblets.. (+4) 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47 has 37 jerblets.. (+4)

1,3,5,9,11,14,18,20,24,29,33,37... sequence isn't in OEIS)

1,2,2,4,2,3,4,2,4,5,4,4,... sequence IS in OEIS!!)

"Number of ways of writing prime(n) in the form 2*prime(i)+prime(j)"

Thats a cool result, jerblets are related to primes I think.

The jerblet definition is z=(y-x)/2 where x,y,z are all in the sequence so that is similar to the 2*prime(i)+prime(j) I guess..

Also the jerblet count is close to the max value in the sequence of primes that creates the jerblet count sometimes..

ie:

from this mess below, the max value in the sequence of primes is pretty close to the jerblet count. Not sure if it stays close for really long sequences of primes or not. (will have to check). If jerblet count does track the sequence max value within a percent error, then I would say that is cool cause it means that something is going on that relates the jerblet formula to the sequence of primes, ie the spacing of primes would be such that it is related to the jerblet formula. (just speculation probably not true that they are related like that)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107 has 100 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109 has 106 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113 has 114 jerblets.. (+8)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127 has 120 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131 has 126 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137 has 134 jerblets.. (+8)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139 has 139 jerblets.. (+5)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149 has 147 jerblets.. (+8)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151 has 153 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157 has 159 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157, 163 has 168 jerblets.. (+9)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167 has 173 jerblets.. (+5)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173 has 182 jerblets.. (+9)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179 has 189 jerblets.. (+7)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181 has 195 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191 has 201 jerblets.. (+6)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191,193 has 208 jerblets.. (+7)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191,193,197 has 218 jerblets.. (+10)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191,193,197,199 has 225 jerblets.. (+7)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191,193,197,199,211 has 233 jerblets.. (+8)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191,193,197,199,211,223 has 241 jerblets.. (+8)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,

163,167,173,179,181,191,193,197,199,211,223,227 has 247 jerblets.. (+6)

cheers, Jamie

Just wanted to add that I already came at this problem of jerblets from another direction before, when I had jerblets named "primal pairs".

I was a bit confused for awhile what jerblets were as it is a strange thing but now I see they are just "primal pairs" really, which were due for a renaming anyway.

The "primal pair Z(count)" I was talking about before is just the count of jerblet center points for each given center point.

ie for jerblet 3 7, that has a center point of 5.

For the prime sequence jerblets, if you graph the count of jerblet center points, it has peaks on the primorial numbers 6,30,210,2310, so this is how the jerblets are related to prime numbers.

Here is the graph yet again showing the jerblet center point counts for the prime numbers:

zoomed in: (shows the primorial peaks)

zoomed out: (shows the primorial multiple bands forming)

The zoomed out graph shows how there are areas that are "off limits" to jerblet counts which shows the patterns in the prime numbers.

I should make some of these graphs for sequences besides the primes to see if the graphs can describe patterns in the sequence like how it described the patterns of primorial number peaks in the prime sequence.

I think jerblets (aka primal pairs) and jerblet center point counts are a decent framework for the prime numbers that shows some of how they are organized, and maybe can be used for finding out more about the prime numbers.

cheers, Jamie

Ok I stumbled upon something that should raise the spirits of even the infamous Bill Slowman, as the jerblets are now related to the most special time of year, Christmas, through the 12 days of Christmas song.

ie. the above sequence 1,2,4,6,9,12,16.. shown above here and on OEIS:

has a description on OEIS showing the sequences true holiday spirit meaning:

from OEIS: "1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the n-th day in the "Twelve Days of Christmas" song. For example, on the fifth day of Christmas, you have 9 French hens. - Alonso del Arte, Jun 17 2005 "

from wikipedia:

"The Twelve Days of Christmas" is a cumulative song, meaning that each verse is built on top of the previous verses. There are twelve verses, each describing a gift given by "my true love" on one of the twelve days of Christmas.

On the First day of Christmas my true love sent to me a Partridge in a Pear Tree.

On the Second day of Christmas my true love sent to me Two Turtle Doves and a Partridge in a Pear Tree.

On the Third day of Christmas my true love sent to me Three French Hens,[3] Two Turtle Doves and a Partridge in a Pear Tree.

So yep the jerblets are related to prime numbers and more importantly to Christmas!

That sequence is really amazing how many different patterns people found in it, there are many different meanings the sequence has on that OEIS page, ie for chess:

from the same:

"Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task. - Rick L. Shepherd, Sep 17 2002

For example, a(2) = 1 and one capture can produce "doubled pawns", a(3) = 2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, non-capturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.) - Rick L. Shepherd, Sep 17 2002 "

cheers, Jamie