A045345 -id:A045345 - cp's OEIS frontend

A050247 a(n) is the sum of the first A045345(n) primes.

2, 874, 5830, 2615298, 712377380, 86810649294, 794712005370, 105784534314378, 92542301212047102, 82704567079549985700, 24733255676526572596026, 3612032130800990065125528, 8102184022428756416738378

Offset: 1

Eric W. Weisstein

a(10) and a(11) were found by Giovanni Resta (Nov 15 2004). He states that there are no other terms for primes p < 4011201392413. See link to Prime Puzzles, Puzzle 31 below.
a(13) > 6640510710493148698166596 (sum of first pi(2*10^13) primes). - Donovan Johnson, Aug 23 2010
a(16) > 416714769731839517991408161209 (sum of first pi(1.55*10^14) primes). - Bruce Garner, Mar 06 2021
a(17) > 814043439429001245436559390420866 (sum of first 6500000004150767 primes). - Paul W. Dyson, Sep 27 2022

Paul W. Dyson, Table of n, a(n) for n = 1..16 (terms 1..15 from Bruce Garner).
OEIS Wiki, Sums of powers of primes divisibility sequences
Carlos Rivera, Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).

Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.

A007504 = Cases[Import["https://oeis.org/A007504/b007504.txt", "Table"], {, }][[All, 2]];
A045345 = Cases[Import["https://oeis.org/A045345/b045345.txt", "Table"], {, }][[All, 2]];
Table[A007504[[A045345[[n]] + 1]], {n, 1, 5}](* Robert Price, Mar 13 2020 *)

s=n=0; forprime(p=2, 1e9, if((s+=p)%n++==0, print1(s", "))) \\ Charles R Greathouse IV, Nov 07 2014

a(n) = Sum_{i=1..A045345(n)} A000040(i). - R. J. Mathar, Jan 26 2008

a(10)-a(11) from Giovanni Resta submitted by Ray Chandler, Jul 19 2010
a(12) from Donovan Johnson, Aug 23 2010
a(13) from Robert Price, Mar 17 2013

A007504 Sum of the first n primes.

Original entry on oeis.org

0, 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, 3087, 3266, 3447, 3638, 3831, 4028, 4227, 4438, 4661, 4888

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

It appears that a(n)^2 - a(n-1)^2 = A034960(n). - Gary Detlefs, Dec 20 2011
This is true. Proof: By definition we have A034960(n) = Sum_{k = (a(n-1)+1)..a(n)} (2*k-1). Since Sum_{k = 1..n} (2*k-1) = n^2, it follows A034960(n) = a(n)^2 - a(n-1)^2, for n > 1. - Hieronymus Fischer, Sep 27 2012 [formulas above adjusted to changed offset of A034960 - Hieronymus Fischer, Oct 14 2012]
Row sums of the triangle in A037126. - Reinhard Zumkeller, Oct 01 2012
Ramanujan noticed the apparent identity between the prime parts partition numbers A000607 and the expansion of Sum_{k >= 0} x^a(k)/((1-x)...(1-x^k)), cf. A046676. See A192541 for the difference between the two. - M. F. Hasler, Mar 05 2014
For n > 0: row 1 in A254858. - Reinhard Zumkeller, Feb 08 2015
a(n) is the smallest number that can be partitioned into n distinct primes. - Alonso del Arte, May 30 2017
For a(n) < m < a(n+1), n > 0, at least one m is a perfect square.
Proof: For n = 1, 2, ..., 6, the proposition is clear. For n > 6, a(n) < ((prime(n) - 1)/2)^2, set (k - 1)^2 <= a(n) < k^2 < ((prime(n) + 1)/2)^2, then k^2 < (k - 1)^2 + prime(n) <= a(n) + prime(n) = a(n+1), so m = k^2 is this perfect square. - Jinyuan Wang, Oct 04 2018
For n >= 5 we have a(n) < ((prime(n)+1)/2)^2. This can be shown by noting that ((prime(n)+1)/2)^2 - ((prime(n-1)+1)/2)^2 - prime(n) = (prime(n)+prime(n-1))*(prime(n)-prime(n-1)-2)/4 >= 0. - Jianing Song, Nov 13 2022
Washington gives an oscillation formula for |a(n) - pi(n^2)|, see links. - Charles R Greathouse IV, Dec 07 2022

E. Bach and J. Shallit, §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms, MIT Press, Cambridge, MA, 1996.
H. L. Nelson, "Prime Sums", J. Rec. Math., 14 (1981), 205-206.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 0..100000
C. Axler, On a Sequence involving Prime Numbers, J. Int. Seq. 18 (2015) # 15.7.6.
Christian Axler, New bounds for the sum of the first n prime numbers, arXiv:1606.06874 [math.NT], 2016.
P. Hecht, Post-Quantum Cryptography: S_381 Cyclic Subgroup of High Order, International Journal of Advanced Engineering Research and Science (IJAERS, 2017) Vol. 4, Issue 6, 78-86.
R. J. Mathar, Table of 100000n, a(100000n) for n = 1..10000
Romeo Meštrović, Curious conjectures on the distribution of primes among the sums of the first 2n primes, arXiv:1804.04198 [math.NT], 2018.
Vladimir Shevelev, Asymptotics of sum of the first n primes with a remainder term
Nilotpal Kanti Sinha, On the asymptotic expansion of the sum of the first n primes, arXiv:1011.1667 [math.NT], 2010-2015.
Lawrence C. Washington, Sums of Powers of Primes II, arXiv preprint (2022). arXiv:2209.12845 [math.NT]
Eric Weisstein's World of Mathematics, Prime Sums
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A000041, A034386, A111287, A013916, A013918 (primes), A045345, A050247, A050248, A068873, A073619, A034387, A014148, A014150, A178138, A254784, A254858.
See A122989 for the value of Sum_{n >= 1} 1/a(n).
Cf. A008472, A002110, A038107.

P:=Filtered([1..250],IsPrime);;
a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # Muniru A Asiru, Oct 07 2018

a007504 n = a007504_list !! n
a007504_list = scanl (+) 0 a000040_list
-- Reinhard Zumkeller, Oct 01 2014, Oct 03 2011

[0] cat [&+[ NthPrime(k): k in [1..n]]: n in [1..50]]; // Bruno Berselli, Apr 11 2011 (adapted by Vincenzo Librandi, Nov 27 2015 after Hasler's change on Mar 05 2014)

s1:=[2]; for n from 2 to 1000 do s1:=[op(s1),s1[n-1]+ithprime(n)]; od: s1;
A007504 := proc(n)
    add(ithprime(i), i=1..n) ;
end proc: # R. J. Mathar, Sep 20 2015

Accumulate[Prime[Range[100]]] (* Zak Seidov, Apr 10 2011 *)
primeRunSum = 0; Table[primeRunSum = primeRunSum + Prime[k], {k, 100}] (* Zak Seidov, Apr 16 2011 *)

A007504(n) = sum(k=1,n,prime(k)) \\ Michael B. Porter, Feb 26 2010

a(n) = vecsum(primes(n)); \\ Michel Marcus, Feb 06 2021

from itertools import accumulate, count, islice
from sympy import prime
def A007504_gen(): return accumulate(prime(n) if n > 0 else 0 for n in count(0))
A007504_list = list(islice(A007504_gen(),20)) # Chai Wah Wu, Feb 23 2022

a(n) ~ n^2 * log(n) / 2. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 24 2001 (see Bach & Shallit 1996)
a(n) = A014284(n+1) - 1. - Jaroslav Krizek, Aug 19 2009
a(n+1) - a(n) = A000040(n+1). - Jaroslav Krizek, Aug 19 2009
a(A051838(n)) = A002110(A051838(n)) / A116536(n). - Reinhard Zumkeller, Oct 03 2011
a(n) = min(A068873(n), A073619(n)) for n > 1. - Jonathan Sondow, Jul 10 2012
a(n) = A033286(n) - A152535(n). - Omar E. Pol, Aug 09 2012
For n >= 3, a(n) >= (n-1)^2 * (log(n-1) - 1/2)/2 and a(n) <= n*(n+1)*(log(n) + log(log(n))+ 1)/2. Thus a(n) = n^2 * log(n) / 2 + O(n^2*log(log(n))). It is more precise than in Fares's comment. - Vladimir Shevelev, Aug 01 2013
a(n) = (n^2/2)*(log n + log log n - 3/2 + (log log n - 3)/log n + (2 (log log n)^2 - 14 log log n + 27)/(4 log^2 n) + O((log log n/log n)^3)) [Sinha]. - Charles R Greathouse IV, Jun 11 2015
G.f: (x*b(x))/(1-x), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 10 2016
a(n) = A008472(A002110(n)), for n > 0. - Michel Marcus, Jul 16 2020

More terms from Stefan Steinerberger, Apr 11 2006

a(0) = 0 prepended by M. F. Hasler, Mar 05 2014

A050248 Integer averages of first k primes for some k.

Original entry on oeis.org

2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, 31043311588, 569424748566, 7207204117608, 10871205353578, 196523412770096, 2665506690112870, 122498079071529726

Offset: 1

Eric W. Weisstein

a(10) and a(11) were found by Giovanni Resta (Nov 15 2004). He states that there are no other terms for primes p < 4011201392413. See link to Prime Puzzles, Puzzle 31 below.
a(13) > (sum of first pi(2*10^13) primes)/pi(2*10^13). - Donovan Johnson, Aug 23 2010
a(16) > 2688482385366706. - Bruce Garner, Mar 06 2021
a(17) > 125237452139872271. - Paul W. Dyson, Sep 26 2022

			38 is average of first k = 23 primes; 110 (k = 53); 3066 (k = 853); 60020 (k = 11869).

OEIS Wiki, Sums of powers of primes divisibility sequences
Carlos Rivera, Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Prime Sums.

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

a=0;lst={};Do[p=Prime[n];a+=p;If[a/n==IntegerPart[a/n],AppendTo[lst,a/n]],{n,10!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 05 2009 *)
Module[{nn=10^6,prs},prs=Prime[Range[nn]];Select[Table[Mean[Take[prs,n]],{n,nn}],IntegerQ]] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Jun 12 2024 *)

s=n=0;forprime(p=2,1e9,if((s+=p)%n++==0, print1(s/n", "))) \\ Charles R Greathouse IV, Nov 07 2014

a(n) = A050247(n)/A045345(n).

Edited by N. J. A. Sloane at the suggestion of David W. Wilson, Jun 23 2007
a(10)-a(11) from Giovanni Resta via Ray Chandler, Jul 19 2010
a(12) from Donovan Johnson, Aug 23 2010
a(13) from Robert Price, Mar 17 2013
a(14)-a(15) from Bruce Garner, Mar 06 2021
a(16) from Paul W. Dyson, Sep 26 2022

A111441 Numbers k such that the sum of the squares of the first k primes is divisible by k.

Original entry on oeis.org

1, 19, 37, 455, 509, 575, 20597, 202717, 1864637, 542474231, 1139733677, 51283502951, 230026580777, 22148897608321, 51271840444039, 1820988137264459

Offset: 1

Stefan Steinerberger, Nov 14 2005

a(16) > 10^14 if it exists. - Anders Kaseorg, Dec 02 2020
Conjecture: There are no terms that are 3 or 9 modulo 12. This seems to hold for all related sequences with even powers of primes, not just squares. Compare "sums of powers of primes divisibility sequences", linked below. - Daniel Bamberger, Dec 03 2020
From Jacob Christian Munch-Andersen, Dec 13 2020: (Start)
Any prime except 3 raised to the 2nd power is 1 modulo 3. Therefore adding the squared primes together results in a simple periodic pattern modulo 3. Any term that is 0 modulo 3 would imply that it divides a number that is 2 modulo 3; as this is impossible there cannot be any terms divisible by 3.
The same proof indeed holds for similar lists generated with any even power, and a similar proof for instance disqualifies any multiple of 5 from the similar 4th-power list. A slightly simpler similar proof shows that there are no terms divisible by 2.
(End)
The previous comment implies that for a list generated with the m-th power, there are no terms divisible by p when p is prime and p-1 is a divisor of m. For example, the 12th power list has no terms divisible by 2, 3, 5, 7 or 13. - Paul W. Dyson, Jan 09 2021
The periodic pattern of the sum of primes raised to an even power as described in the comments above follows from Fermat's little theorem. When the pattern is periodic for a given p it can be seen that when k mod p = 0 the sum mod p = p-1 and therefore sum mod k cannot be 0. - Bruce Garner, Apr 08 2021
a(2) is also a value in each of the lists generated with the powers 20, 38, 56... . a(3) is also a value in each of the lists generated with the powers 38, 74, 110... . In general, if the sum of the first k primes each to the power of m is divisible by k, and m >= the maximum exponent in the prime factorization of k, then the sum of the first k primes each to the power of m + j * psi(k) is also divisible by k, where psi(k) is the reduced totient function (A002322) and j is any positive integer. This follows from the fact that n^m == n^(m + psi(k)) (mod k) for all integers n and all integers m >= the maximum exponent in the prime factorization of k. - Paul W. Dyson, Dec 09 2022
a(17) > 8*10^15. - Paul W. Dyson, Jan 16 2025

			The sum of the squares of the first 19 primes 2^2 + 3^2 + 5^2 + ... + 67^2 = 19*1314, thus 19 is in the sequence.

OEIS Wiki, Sums of powers of primes divisibility sequences.
Matt Parker, MPMSolutions: The 19 Challenge, YouTube video, 2020 (challenge asking for terms of this sequence).
Wikipedia, Fermat's little theorem

Cf. A024450, A045345.

Cf. also A217599, A217600 for the corresponding prime numbers and sums.

s = 0; t = {}; Do[s = s + Prime[n]^2; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 10^6}]; t (* Robert G. Wilson v, Nov 15 2005 *)
Module[{nn=2*10^6,pr2},pr2=Accumulate[Prime[Range[nn]]^2];Select[Thread[{Range[nn],pr2}],Divisible[#[[2]],#[[1]]]&]][[;;,1]] (* The program generates the first 9 terms of the sequence. *) (* Harvey P. Dale, May 25 2025 *)

a := 0; for n from 1 to 100000 do a := a + ithprime(n)^2; if a/n = trunc(a/n) then print(n); end_if; end_for;

for(n=1, 2*10^11, m=n; s=0; while(m>0, s=s+prime(m)^2; m--); if(s%n==0, print1(n, ", "))) \\ Felix Fröhlich, Jul 07 2014

isok(n) = norml2(primes(n)) % n == 0; \\ Michel Marcus, Nov 25 2020

a(8)-a(9) from Robert G. Wilson v, Nov 15 2005
a(10)-a(11) from Ryan Propper, Mar 27 2007
a(12) from Robert Price, Mar 19 2013
a(13) from Balázs Dura-Kovács, Nov 25 2020
a(14) from Balázs Dura-Kovács, Nov 30 2020
a(15) from Anders Kaseorg, Dec 02 2020
a(16) from Jonas Lippuner, Aug 23 2021

A171399 Prime(k), where k is such that (Sum_{i=1..k} prime(i)) / k is an integer.

Original entry on oeis.org

2, 83, 241, 6599, 126551, 1544479, 4864121, 60686737, 1966194317, 63481708607, 1161468891953, 14674403807731, 22128836547913, 399379081448429, 5410229663058299, 248264241666057167

Offset: 1

Jaroslav Krizek, Dec 07 2009

Corresponding values of k, Sum_{i=1..k} p_i, and (Sum_{i=1..k} p_i) / k are given in A045345, A050247 and A050248. No other solutions for p_k < 4011201392413.
a(13) > 2*10^13. - Donovan Johnson, Aug 23 2010
a(16) > 5456843462009647. - Bruce Garner, Mar 06 2021
a(17) > 253814097223614463. - Paul W. Dyson, Sep 26 2022

			83 is the 23rd prime and (Sum_{i=1..23} p_i) / 23 = 874 / 23 = 38 (integer), so 83 is a term.

Kaisa Matomäki, A note on the sum of the first n primes, Quart. J. Math. 61 (2010), pp. 109-115.
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

t = {}; sm = 0; Do[sm = sm + Prime[n]; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* T. D. Noe, Mar 19 2013 *)

s=0; n=0; forprime(p=2, 1e7, s+=p; if(s%n++==0, print1(p", "))) \\ Charles R Greathouse IV, Jun 13 2012

a(n) = A000040(A045345(n)).

a(6) corrected and a(12) from Donovan Johnson, Aug 23 2010
a(13) from Robert Price, Mar 17 2013
a(14)-a(15) from Bruce Garner, Mar 06 2021
a(16) from Paul W. Dyson, Sep 26 2022

A217599 Prime(n), where n is such that (Sum_{i=1..n} prime(i)^2) / n is an integer.

Original entry on oeis.org

2, 67, 157, 3217, 3637, 4201, 231947, 2790569, 30116309, 12021325961, 26144296151, 1380187561637, 6549419699279, 735325088697473, 1746583001138813, 68725636353488501

Offset: 1

Robert Price, Mar 19 2013

a(16) > 3*10^15 if it exists. - Anders Kaseorg, Dec 02 2020

a(17) > 3.1*10^17. - Paul W. Dyson, Jan 16 2025

			a(2) = 67, because 67 is the 19th prime and the sum of the first 19 primes^2 = 24966 when divided by 19 equals 1314 which is an integer.

OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.

Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.

t = {}; sm = 0; Do[sm = sm + Prime[n]^2; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* T. D. Noe, Mar 19 2013 *)
k = 1; p = 2; s = 0; lst = {}; While[p < 1000000000, s = s + p^2; If[ Mod[s, k++] == 0, AppendTo[lst, p]]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Mar 08 2015 *)

n=s=0; forprime(p=2,1e9, if((s+=p^2)%n++==0, print1(p", "))) \\ Charles R Greathouse IV, Feb 06 2015

a(13) from Willem Hengeveld, Nov 29 2020
a(14)-a(15) from Anders Kaseorg, Dec 02 2020
a(16) from Paul W. Dyson, Sep 03 2022

A128165 Numbers k such that k divides 1 plus the sum of the first k primes.

Original entry on oeis.org

1, 2, 6, 10, 20, 22, 28, 155, 488, 664, 992, 6162, 7840, 7975, 8793, 18961, 32422, 148220, 231625, 332198, 459121, 462932, 2115894, 8108930, 10336641, 11789731, 15500046, 23483195, 46571611, 48582404, 77033887, 105390951, 132421841, 229481560, 1224959312

Offset: 1

Alexander Adamchuk, Feb 22 2007

a(44) > 4.4*10^10. - Robert Price, Dec 15 2013

a(50) > 10^14. - Bruce Garner, Jun 05 2021

Bruce Garner, Table of n, a(n) for n = 1..49
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).

Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.

k = 0; s = 1; p = 2; A128165 = {}; While[k < 247336000, If[Mod[s += p, ++k] == 0, AppendTo[A128165, k]; Print[{k, p}]]; p = NextPrime@ p]; A128165

is(n)=sum(i=1,n,prime(i),1)%n==0 \\ Charles R Greathouse IV, Nov 07 2014

n=0; s=1; forprime(p=2,1e9, s+=p; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Nov 07 2014

More terms from Ryan Propper, Apr 05 2007
a(34) from Robert G. Wilson v, Jan 21 2011
a(35) from Robert Price, Dec 15 2013

A217600 Sum of the squares of the first A111441(n) primes.

Original entry on oeis.org

4, 24966, 263736, 1401992410, 2040870112, 3054955450, 346739122490032, 499159078330000800, 539391065522650998496, 25318239660367402306502991202, 251882074412384639674100925616, 31734804589156174948658730855096778, 3209990334856119248883461357431048564, 3910080232300154696097509520638192488259772

Offset: 1

Robert Price, Mar 19 2013

nonn

a(n) - A111441(a(n)) - 11 == 0 (mod 24) for n > 1. This is similar to the relation between A000027 and A024450. - Karl-Heinz Hofmann, Jan 11 2021

Paul W. Dyson, Table of n, a(n) for n = 1..16 (terms 1..12 from Robert Price, 13 from Willem Hengeveld, 14..15 from Bruce Garner).
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).

Cf. also A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.

n=s=0; forprime(p=2,1e8, s+=p^2; if(s%n++==0, print1(s", "))) \\ Charles R Greathouse IV, Apr 22 2015

a(13) from Willem Hengeveld, Nov 29 2020
a(14) from Bruce Garner, Dec 16 2020
a(15) from Bruce Garner, Dec 24 2020

A085450 a(n) is the smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.

Original entry on oeis.org

23, 19, 25, 2951, 25, 4357, 25, 43, 281525, 269, 25, 37, 23, 295, 17, 3131, 395191, 37, 25, 19, 139, 1981, 23, 37, 25, 455, 25, 41, 124403, 61, 17, 511, 193, 535, 23, 5209, 1951, 19, 25, 301, 891, 9805, 25, 527, 23, 83, 17, 37, 131, 43, 25, 193, 53, 37, 25, 19

Offset: 1

Farideh Firoozbakht, Aug 14 2003

nonn

By definition a(1) is A045345(2).
This sequence has a very interesting behavior. If Mod(n, 2)(Mod(n, 20)-1)(Mod(n, 20)-9)(Mod(n, 20)-13)(Mod(n, 20)-17)!=0, a(n)=17, 23 or 25; in other cases a(n) may be too large. If Mod[n, 16] = 15, a(n) = 17. For example, a(n) = 17 for n = 15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, ...; also, a(n) = 23 for n = 1, 13, 23, 35, 45, 57, 67, 89, 101, 123, 133, 145, 155, 167, 177, 189, 199, ...; a(n) = 25 for n = 3, 5, 7, 11, 19, 25, 27, 39, 43, 51, 55, 59, 65, 71, 75, ..., . For a(n) = 19 for n = 2, 20, 38, 56, 74, 92, 110, 128, 146, 164, 182, 200, 218, ..., == 2 (mod 18).
From Alexander Adamchuk, Jul 20 2008: (Start)
Conjectures:
a(n) exists for all n; a(n) >= 17.
a(325)-a(575) = {25,19,25,5851,1843,61,23,821,89,301,17,37,131,455,25,1607,297,37,23,19,25,
325,25,37,353,47,17,1663,23,691,25,691,509269,155,25,269,105893,19,25,3971,
23,213215,17,26021,327,79,25,37,151,83,23,161,101,37,25,19,327,265,17,37,25,
43,23,41,169,61,25,113,21761,6289,25,47,23,19,17,4073,1137,565,25,527,25,
325,25,37,23,455,25,431,13195,37,17,19,53,155,23,37,89,455,25,18839,25,6221,
25,41,18597,229,17,811,623173,19,25,193,2079,673,25,881,23,47,25,37,25,97,
17,79,131,37,25,19,23,56501,25,37,299,455,25,167,2707,446963,17,157,25,325,
25,41,53,19,25,5917,103,1051,23,607,101,155,17,37,6233,455,25,9049,23,37,25,
19,327,5359,25,37,43,455,17,9187,23,193,25,1861,7923,301,25,113,25,19,23,41,
89,61,17,43,1785,131,25,37,1417,455,23,151,53,37,25,19,25,79,17,37,23,455,
25,289,59,47,25,511,47,83,25,739,23,19,17,301,25,269,25,41,707,2735,23,37,
299,43,25,283,69723,37,17,19,1785,479,23,37,25,455,25,1867,131,61,25,31799,
23,161,17}.
a(n) is currently unknown and a(n)>10^7 for n = {324, 576, ...}. (End)
All but one of the terms up to n=1000 are known and they are less than 10^8. Currently the only unknown term for n<=1000 is a(656)>10^8. - Alexander Adamchuk, May 24 2009
More terms: a(324) = 18642551, a(576) = 12824827. - Alexander Adamchuk, May 24 2009
a(656) > 23,491,000,000. - Robert Price, Apr 22 2014
a(656) > 10^12. - Paul W. Dyson, Nov 23 2024
From Paul W. Dyson, Jan 18 2025: (Start)
If n == 15 (mod 16), a(n) = 17; otherwise if n == 2 (mod 18), a(n) = 19; otherwise if n mod 22 = 1 or 13, a(n) = 23; otherwise if n mod 20 = 3, 5, 7, 11, 15 or 19, a(n) = 25; otherwise if n mod 36 = 12, 18 or 24, a(n) = 37; etc. These follow from the fact that a(n) will also be a divisor for a prime sum with power j when j == n (mod psi(a(n))) and both n and j are greater than or equal to the maximum exponent in the prime factorization of a(n), where psi is the reduced totient function (A002322). E.g. for n=15, a(n)=17 and psi(a(n)) = 16. So j = 31, 47, 63, ..., and a(31) = a(47) = a(63) = a(15) = 17. For proof, see the comment dated Dec 09 2022 in A111441.
If a(n) exists, a(n) >= 17. For k < 17, psi(k) <= 12 and the maximum exponent in a prime factorization is 4 (as 16=2^4). So any a(n) < 17 would appear with periodicity <= 12, and would be seen in the first 15 (=12+4-1) terms of the sequence. (End)

			a(3) = 25 because 2^3+3^3+5^3...+prime(25)^3 == 0 (mod 25) and for 1 < n < 25 2^3+3^3+...+prime(n)^3 is not congruent to zero (mod n).

Alexander Adamchuk and Robert Price, Table of n, a(n) for n = 1..655 (first 323 terms from Alexander Adamchuk)

Cf. A045345, A111441, A122140, A125907, A122142, A125825, A125826, A125828, A131263, A131264, A125827, A131272, A131273, A131274, A131275, A131276, A131277, A131278, A131279.

a[n_] := Block[{m = 2, s = 2^n}, While[s = s + Prime[m]^n; Mod[s, m] != 0, m++ ]; m]; Table[ a[n], {n, 1, 56}]
a[n_] := Block[{m = 2, s = 2^n}, While[s = s + Prime[m]^n; Mod[s, m] != 0&& m<10000000, m++ ]; m]; Table[ a[n], {n, 1, 1000}] (* Alexander Adamchuk, Jul 20 2008 *)

a(n)=my(s=2^n,m=1); forprime(p=3,, if((s+=p^n)%m++==0, return(m))) \\ Charles R Greathouse IV, Feb 06 2015

For[m=2, Mod[Sum[Prime[k]^n, {k, m}], m]!=0, m++ ]; m

Edited and extended by Robert G. Wilson v, Aug 14 2003

A128166 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^2 = 1 + A024450(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 13, 26, 28, 45, 66, 174, 308, 350, 366, 417, 783, 804, 3774, 5714, 7998, 17628, 17940, 63447, 67620, 83028, 137868, 216717, 297486, 425708, 659316, 674166, 883500, 1203786, 3605052, 6778607, 9516098, 19964862, 25338586, 27771732, 70980884, 91871891, 208234138, 231967260, 238066596, 289829748, 784027092, 1078515812, 33256634230

Offset: 1

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

a(51) > 5.3*10^10. - Robert Price, Dec 16 2013

a(67) > 7*10^13. - Bruce Garner, May 05 2021

Bruce Garner, Table of n, a(n) for n = 1..66
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

s = 1; Do[s = s + Prime[n]^2; If[ Mod[s, n] == 0, Print[n]], {n, 700000}]
(* or *)
Select[Range[10^4], IntegerQ[(1 + Plus@@(Prime[Range[#]]^2))/#] &] (* Alonso del Arte, Jan 20 2011 *)

More terms from Sean A. Irvine, Jan 20 2011

a(45)-a(50) from Robert Price, Dec 16 2013

cp's OEIS Frontend

A050247 a(n) is the sum of the first A045345(n) primes.

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A007504 Sum of the first n primes.

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A050248 Integer averages of first k primes for some k.

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A111441 Numbers k such that the sum of the squares of the first k primes is divisible by k.

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A171399 Prime(k), where k is such that (Sum_{i=1..k} prime(i)) / k is an integer.

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A217599 Prime(n), where n is such that (Sum_{i=1..n} prime(i)^2) / n is an integer.

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