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# Sums of primes divisibility sequences

There are many sequences dealing with the sums of primes and their divisibility. They are summarized here.

The sum of the $i\,$ th power of the first n primes: $a=\sum _{k=1}^{n}p_{k}^{i}\,$ One plus the sum of the $i\,$ th power of the first n primes: $b=1+\sum _{k=1}^{n}p_{k}^{i}\,$ ## Related Sequences

Column headings in the table below:

i                             the power, primes are raised
a                             as defined above
n div a                       n divides a
where n is not divisible by   when n is a term, n is not divisible by any number of the set
p(a,n)                        pi(n) where n divides a
b                             as defined above
n div b                       n divides b
p(b,n)                        pi(n) where n divides b
sum(n div a)                  sum of the primes where n divides a
avg(n div a)                  average of the primes where n divides a

NOGI                          Not of great interest



Sequences and Restrictions
i a n div a where n is not
divisible by
p(a,n) n div b p(b,n) sum(n div a) avg(n div a)
1 A007504 A045345 [ 2 ] A171399 A128165 A233523 A050247 A050248
2 A024450 A111441 [ 2, 3 ] A217599 A128166 A233862 A217600 A217601
3 A098999 A122140 [ 2 ] A223936 A128167 A233863 A223937 A223939
4 A122102 A125907 [ 2, 3, 5 ] A232869 A128168 A233893 A341689 A341690
5 A122103 A122142 [ 2 ] A224083 A128169 A234003 A224084 A224087
6 A236182 A125825 [ 2, 3, 7 ] A232733 A128170 A233041 NOGI NOGI
7 A236209 A125826 [ 2 ] A232865 A128171 A233040 NOGI NOGI
8 A236214 A125828 [ 2, 3, 5 ] A232822 A232823 A232824 NOGI NOGI
9 A236215 A131263 [ 2 ] A232962 A232964 A233042 NOGI NOGI
10 A236216 A131264 [ 2, 3, 11 ] A233132 A233133 A233134 NOGI NOGI
11 A234740 A125827 [ 2 ] A233192 A233193 A233194 NOGI NOGI
12 A236218 A131272 [ 2, 3, 5, 7, 13 ] A233263 A233264 A233265 NOGI NOGI
13 A236221 A131273 [ 2 ] A232770 A233349 A233350 NOGI NOGI
14 A236222 A131274 [ 2, 3 ] A232963 A232966 A233043 NOGI NOGI
15 A236223 A131275 [ 2 ] A232848 A233413 A233414 NOGI NOGI
16 A236224 A131276 [ 2, 3, 5, 17 ] A233460 A233461 A233462 NOGI NOGI
17 A236225 A131277 [ 2 ] A233555 A233556 A233557 NOGI NOGI
18 A236226 A131278 [ 2, 3, 7, 19 ] A233575 A233576 A233577 NOGI NOGI
19 A236227 A131279 [ 2 ] A233767 A233768 A233769 NOGI NOGI
There are no divisibility restrictions known for "n div b". Karl-Heinz Hofmann  Mar 11 2021.