[seqfan] Re: help with a sequence

[Bob Selcoe]

Hi Jamie,

Ok, if I understand correctly, what you're saying is when (prime(i) - prime 
(j))/2 is prime, then the most ways to express n = (prime(i) + prime(j))/2 
is when n = primorial(k)?  But that doesn't hold for 6 and 30.  And it seems 
like there are a lot of counter-examples for a(n) being relatively large 
when n has a relatively large number of unique primes.

Do you mean, rather, that there tend to be peaks when n is a multiple of 
primorial(k), and the peaks amplify as k increases?  The first graph 
certainly suggests that may be the case.  If so, I would definitely 
recommend adding your sequence {0,0,0,0,0,1,0,0,2,1,2,0,2...} i.e., a(n) is 
the number of ways to express n = (prime(i) + prime(j))/2 when (prime(i) - 
prime (j))/2 also is prime.  (Perhaps there's a better name for it).

BTW - what is the next prime n after a(5) where a(n) > 0?

Cheers,
Bob S
____


> > Now if you ask the question how many of these equations does a given
> centerpoint have, then that is what the significance would be since it 
> shows
> that there are peaks in the centerpoint count on numbers that have the 
> most
> unique prime factors ie primorials.
>
> For example the centerpoint 8 has 2 valid equations using primes for N:
> (11-5)/2=3
> (13-3)/2=5
>
> ie. here is the count of how many equations all the centerpoints from 0 to 
> 216 have, and
> centerpoint 210 a primorial has the most equations.
>
> 0    0
> 1    0
> 2    0
> 3    0
> 4    0
> 5    1
> 6    0
> 7    0
> 8    2
> 9    1
> 10    2
> 11    0
> 12    2
> 13    0
> 14    1
> 15    1
> 16    2
> 17    0
> 18    3
> 19    0
> 20    2
> 21    1
> 22    1
> 23    0
> 24    5
> 25    0
> 26    1
> 27    0
> 28    0
> 29    0
> 30    5
> 31    0
> 32    1
> 33    0
> 34    1
> 35    0
> 36    5
> 37    0
> 38    0
> 39    1
> 40    1
> 41    0
> 42    6
> 43    0
> 44    1
> 45    1
> 46    1
> 47    0
> 48    5
> 49    0
> 50    2
> 51    0
> 52    0
> 53    0
> 54    5
> 55    0
> 56    2
> 57    0
> 58    0
> 59    0
> 60    10
> 61    0
> 62    0
> 63    0
> 64    1
> 65    0
> 66    8
> 67    0
> 68    0
> 69    1
> 70    2
> 71    0
> 72    6
> 73    0
> 74    0
> 75    0
> 76    2
> 77    0
> 78    8
> 79    0
> 80    0
> 81    1
> 82    0
> 83    0
> 84    10
> 85    0
> 86    1
> 87    0
> 88    0
> 89    0
> 90    12
> 91    0
> 92    1
> 93    0
> 94    0
> 95    0
> 96    7
> 97    0
> 98    0
> 99    1
> 100    2
> 101    0
> 102    7
> 103    0
> 104    1
> 105    1
> 106    1
> 107    0
> 108    7
> 109    0
> 110    1
> 111    1
> 112    0
> 113    0
> 114    8
> 115    0
> 116    1
> 117    0
> 118    0
> 119    0
> 120    16
> 121    0
> 122    0
> 123    0
> 124    0
> 125    0
> 126    11
> 127    0
> 128    0
> 129    1
> 130    1
> 131    0
> 132    8
> 133    0
> 134    1
> 135    0
> 136    0
> 137    0
> 138    6
> 139    0
> 140    1
> 141    0
> 142    1
> 143    0
> 144    12
> 145    0
> 146    0
> 147    0
> 148    0
> 149    0
> 150    13
> 151    0
> 152    0
> 153    0
> 154    1
> 155    0
> 156    10
> 157    0
> 158    0
> 159    0
> 160    2
> 161    0
> 162    11
> 163    0
> 164    0
> 165    1
> 166    0
> 167    0
> 168    13
> 169    0
> 170    2
> 171    0
> 172    0
> 173    0
> 174    10
> 175    0
> 176    2
> 177    0
> 178    0
> 179    0
> 180    16
> 181    0
> 182    0
> 183    0
> 184    0
> 185    0
> 186    13
> 187    0
> 188    0
> 189    0
> 190    0
> 191    0
> 192    6
> 193    0
> 194    1
> 195    1
> 196    2
> 197    0
> 198    9
> 199    0
> 200    1
> 201    0
> 202    1
> 203    0
> 204    12
> 205    0
> 206    0
> 207    0
> 208    0
> 209    0
> 210    26
> 211    0
> 212    0
> 213    0
> 214    0
> 215    0
> 216    9
>
>
> Here is a graph showing this too:
>
> zoomed in: (shows the primorial peaks)
> http://imgur.com/x7yDW1e
>
> zoomed out: (shows the primorial multiple bands forming)
> http://imgur.com/gcw7S39
>
>
> cheers,
> Jamie
>
>