A hierarchy of double, quadruple and octuple primes

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Primality of q + 26, q + 28, q + 32 and q + 34 for general q-center=q_

q-center	q + 26	q + 28	q + 32	q + 34
195	NO	YES	YES	YES
825	NO	YES	YES	YES
1485	YES	NO	NO	NO
1875	NO	YES	YES	NO
2085	YES	YES	NO	NO
21015	NO	NO	NO	NO

The octuple with o-center= o = 1006320_

type of center	center label	center value	factored center
d-center	d11	1006302	(2,1),(3,1),(11,1),(79,1),(193,1)
q-center	q1	1006305	(3,1),(5,1),(73,1),(919,1)
d-center	d12	1006308	(2,2),(3,2),(27953,1)
o-center	o	1006320	(2,4),(3,1),(5,1),(7,1),(599,1)
d-center	d21	1006332	(2,2),(3,1),(17,1),(4933,1)
q-center	q2	1006335	(3,2),(5,1),(11,1),(19,1),(107,1)
d-center	d22	1006338	(2,1),(3,1),(179,1),(937,1)

Primality o − 1 and o + 1 for select cases of octuple primes_

o =o-center	prime o − 1	prime o + 1
1006320	False	False
2594970	False	True
3919230	True	False
9600570	False	True
10531080	False	False
157131660	False	True
179028780	True	False
211950270	True	False
255352230	False	False
267587880	False	False
724491390	True	False
871411380	False	False

Pure and impure octuples up to 1011_

Number of octuple primes	Pure octuple primes	Octuple primes with o − 1 prime	Octuple primes with o + 1 prime	Octuple primes with o − 1 and o + 1 prime
267	187	34	47	3

Pure and impure octuples up to 1010_

Number of octuple primes	Pure octuple primes	Octuple prime with o − 1 prime	Ocuple primes with o + 1 prime	Ocuple primes with o − 1 and o + 1 prime
65	42	9	14	0

Divisors in the neighborhood of the first o-center o of a possible sixtentuple prime_

index	qn−1 − 4	qn−1 − 3	qn−1 − 2	qn−1 − 1	qn−1	qn−1 + 1	qn−1 + 2	qn−1 + 3	qn−1 + 4
divis	unsp.	2,3	unsp.	2	3,5	2,7	unsp.	2,3	unsp.

Divisibility by 3_

index

q − 9

q − 8

q − 7

q − 6

q − 5

q − 4

q − 3

q − 2

q − 1

q

q + 1

q + 2

q + 3

q+4

divis

2,3

2

3

2

P

2,3

P

2

3

2

P

2,3

P

Divisibility by 2, 3 and 5 around qprime q_

index

q − 7

q − 6

q − 5

q − 4

q − 3

q − 2

q − 1

q

q + 1

q + 2

q + 3

q + 4

divis

2

3

2,5

P

2,3

P

2

3,5

2

P

2,3

P

Divisors in the neighborhood of the quadruple prime with qcenter 195, now including 7_

index	185	186	187	188	189	190	191	192	193	194
divis	5	2,3		2	3,7	2,5	P	2,3	P	2
index	195	196	197	198	199	200	201	202	203	204
divis	3,5	2,7	P	2,3	P	2,5	3	2	7	2,3
index	205	206	207	208	209	210	211	212	213	214
divis	5	2	3	2		2,3,5,7	P	2	3	2
index	215	216	217	218	219	220	221	222	223	224
divis	5	2,3	7	2	3	2,5		2,3	P	2,7
index	225	226	227	228	229	230	231	232	233	234
divis	3,5	2	P	2,3	P	2,5	3,7	2	P	E,3
index	235	236	237	238	239	240	241	242	243	244
divis	5	2	3	2,7	P	2,3,5	P	2	3	2

Divisors in the neighborhood of the second o-center on of a possible sixtentuple prime_

index	qn+1 − 4	qn+1 − 3	qn+1 − 2	qn+1 − 1	qn+1	qn+1 + 1	qn+1 + 2	qn+1 + 3	qn+1 + 4
divis	unsp.	2,3	unsp.	2,7	3,5	2	unsp.	2,3	unsp.

Quadruple prime example_

index	10	11	12	13	14	15	16	17	18	19	20
divis	E	P	d1	P	E	q-center	E	P	d2	P	E

Divisors of centers_

	centers divisible by	span divisible by	comment
dcenter	1	1*2	verified
qcenter	3	1*2*3	verified
ocenter	15	1*2*3*5	verified
scenter	105	1*2*3*5*7	hypothetical

Double prime example_

index	d − 3	d − 2	d − 1	d	d + 1	d + 2	d + 3	d + 4	d + 5
divis		E	P	dcenter	P	E		E