[seqfan] Re: help with a sequence
Jamie Morken
jmorken at shaw.ca
Thu Aug 4 13:44:52 CEST 2016
Hi,
Yep you are right it is quarter-squares if N is all odd numbers, even numbers, or all positive integers too.
For your question about if Z has any significance, ie for the sequence N = 2,3,5,7,11
Z=3
(7-3)/2=2
(11-5)/2=3
(11-7)/2=2
The significance is related to something else I was working on, ie for each of those three
equations if you replace the minus sign with a plus sign then you get this:
(7+3)/2=5
(11+5)/2=8
(11+7)/2=9
So 5,8,9 are the centerpoints between 7,3 and 11,5 and 11,7 respectively.
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
----- Original Message -----
From: Bob Selcoe
rselcoe at entouchonline.net
Tue Aug 2 12:46:08 CEST 2016
Subject: [seqfan] Re: help with a sequenceHi Jamie & Seqfans,
It's an interesting idea; I could certainly see adding, say, the sequence
for N as a set of prime numbers. But I'm not sure why you might want to
apply this particular construct to existing sequences. Why not, for
example, use (y-x)/3 instead?? Does Z have particular significance??
BTW - unless I'm mistaken, we also obtain quarter-squares if N is all
positive integers. I suppose that could be applied to Oppermann also, but
it seems a bit trivial to me.
Cheers,
Bob Selcoe
--------------------------------------------------
From: "Jamie Morken" <jmorken at shaw.ca>
Sent: Monday, August 01, 2016 11:34 PM
To: <seqfan at list.seqfan.eu>
Subject: [seqfan] help with a sequence
> Hi,
>
> Please excuse my lack of correct math terminology, I have a couple
> questions
> about some sequences that I found, and noticed they are already are in
> OEIS,
> but I don't think they are described with this formula, so if it makes
> sense please add it
> to OEIS, or please explain it more to me.
>
> For a partial set of the positive integers called N, for all combinations
> of pairs of integers x and y
> in the set where y is larger than x.
>
> Take the total count of unique formulas (y-x) / 2 = C where C is also in N
> and call this count Z.
>
>
> example with N as sets of prime numbers:
>
> 2,3 has Z=0..
> 2,3,5 has Z=0.. (+0)
> 2,3,5,7 has Z=1.. (+1)
> 2,3,5,7,11 has Z=3.. (+2)
> 2,3,5,7,11,13 has Z=5.. (+2)
> 2,3,5,7,11,13,17 has Z=9.. (+4)
> 2,3,5,7,11,13,17,19 has Z=11.. (+2)
> 2,3,5,7,11,13,17,19,23 has Z=14.. (+3)
> 2,3,5,7,11,13,17,19,23,29 has Z=18.. (+4)
> 2,3,5,7,11,13,17,19,23,29,31 has Z=20.. (+2)
> 2,3,5,7,11,13,17,19,23,29,31,37 has Z=24.. (+4)
> 2,3,5,7,11,13,17,19,23,29,31,37,41 has Z=29.. (+5)
> 2,3,5,7,11,13,17,19,23,29,31,37,41,43 has Z=33.. (+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 http://oeis.org/A103274
> "Number of ways of writing prime(n) in the form 2*prime(i)+prime(j)"
>
>
> example with N as sets of odd numbers:
>
> 3,5,7 has Z=0..
> 3,5,7,9 has Z=1..(+1)
> 3,5,7,9,11 has Z=2..(+1)
> 3,5,7,9,11,13 has Z=4..(+2)
> 3,5,7,9,11,13,15 has Z=6..(+2)
> 3,5,7,9,11,13,15,17 has Z=9..(+3)
> 3,5,7,9,11,13,15,17,19 has Z=12..(+3)
> 3,5,7,9,11,13,15,17,19,21 has Z=16..(+4)
> 3,5,7,9,11,13,15,17,19,21,23 has Z=20..(+4)
> 3,5,7,9,11,13,15,17,19,21,23,25 has Z=25..(+5)
> 3,5,7,9,11,13,15,17,19,21,23,25,27 has Z=30..(+5)
> 3,5,7,9,11,13,15,17,19,21,23,25,27,29 has Z=36..(+6)
>
> 1,2,4,6,9,12,16,20,25,30,36, is in oeis:
>
> Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4)
> http://oeis.org/A002620
>
> 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
> http://oeis.org/A087811
>
> Thanks.
>
> cheers,
> Jamie
>
>
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>
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