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 $\scriptstyle i\,$th power of the first n primes: $a=\sum _{k=1}^{n}p_{k}^{i}\,$

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

Related Sequences

Column headings in the table below:

a              as defined above
n div a        n divides a
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

i	a	n div a	p(a,n)	n div b	p(b,n)	sum(n div a)	avg(n div a)
1	A007504	A045345	A171399	A128165	A233523	A050247	A050248
2	A024450	A111441	A217599	A128166	A233862	A217600	A217601
3	A098999	A122140	A223936	A128167	A233863	A223937	A223939
4	A122102	A125907	A232869	A128168	A233893
5	A122103	A122142	A224083	A128169	A234003	A224084	A224087
6	A236182	A125825	A232733	A128170	A233041	NOGI	NOGI
7	A236209	A125826	A232865	A128171	A233040	NOGI	NOGI
8	A236214	A125828	A232822	A232823	A232824	NOGI	NOGI
9	A236215	A131263	A232962	A232964	A233042	NOGI	NOGI
10	A236216	A131264	A233132	A233133	A233134	NOGI	NOGI
11	A234740	A125827	A233192	A233193	A233194	NOGI	NOGI
12	A236218	A131272	A233263	A233264	A233265	NOGI	NOGI
13	A236221	A131273	A232770	A233349	A233350	NOGI	NOGI
14	A236222	A131274	A232963	A232966	A233043	NOGI	NOGI
15	A236223	A131275	A232848	A233413	A233414	NOGI	NOGI
16	A236224	A131276	A233460	A233461	A233462	NOGI	NOGI
17	A236225	A131277	A233555	A233556	A233557	NOGI	NOGI
18	A236226	A131278	A233575	A233576	A233577	NOGI	NOGI
19	A236227	A131279	A233767	A233768	A233769	NOGI	NOGI

Links

• Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.

• Kaisa Matomäki, A note on the sum of the first n primes, Quart. J. Math. 61 (2010), pp. 109-115.

• The Prime Puzzles & Problems Connection: Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k.

• Eric Weisstein's World of Mathematics, Prime Sums