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:

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.

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