A024450 -id:A024450 - cp's OEIS frontend

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

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

A217601 Integer averages of squares of first primes.

Original entry on oeis.org

4, 1314, 7128, 3081302, 4009568, 5312966, 16834447856, 2462344442400, 289274033242208, 46671783125431818542, 221000817555367050608, 618811172463743796896678, 13954866972387224169218132, 176536110349401666017009273532, 996528450408723697487070591774

Offset: 1

Robert Price, Mar 19 2013

nonn

			a(2) = 1314 is the average of squares of first 19 primes (24966/19=1314).

Paul W. Dyson, Table of n, a(n) for n = 1..16
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.

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

a(n) = A217600(n)/A111441(n).

a(13) from Karl-Heinz Hofmann, Dec 08 2020
a(14) from Karl-Heinz Hofmann, Dec 26 2020
a(15) from Karl-Heinz Hofmann, Dec 27 2020

A233523 Prime(n), where n is such that (1+sum_{i=1..n} prime(i)) / n is an integer.

Original entry on oeis.org

2, 3, 13, 29, 71, 79, 107, 907, 3491, 4967, 7853, 61223, 80051, 81547, 90901, 211811, 381629, 1990007, 3220793, 4749637, 6725027, 6784937, 34463699, 143691323, 185831033, 213609173, 285336497, 442634651, 911588849, 953122843, 1548789581, 2153787017

Offset: 1

Robert Price, Dec 15 2013

nonn

a(50) > 3475385758524527. - Bruce Garner, Jun 05 2021

			a(3) = 13, because 13 is the 6th prime and the sum of the first 6 primes+1 = 42 when divided by 6 equals 7 which is an integer.

Bruce Garner, Table of n, a(n) for n = 1..49 (first 43 terms from Robert Price)
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 = 1; Do[sm = sm + Prime[n]; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233862 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^2) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 37, 41, 101, 107, 197, 317, 1033, 2029, 2357, 2473, 2879, 5987, 6173, 35437, 56369, 81769, 195691, 199457, 793187, 850027, 1062931, 1840453, 2998421, 4217771, 6200923, 9914351, 10153807, 13563889, 18878099, 60767923, 118825361, 170244929

Offset: 1

Robert Price, Dec 16 2013

nonn

a(51) > 1428199016921.

a(67) > 2407033812270611. - Bruce Garner, May 05 2021

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^2+1 = 378 when divided by 6 equals 63 which is an integer.

Bruce Garner, Table of n, a(n) for n = 1..66 (first 50 terms from Robert Price)
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 = 1; Do[sm = sm + Prime[n]^2; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=9600000},Prime[#]&/@Transpose[Select[Thread[{Range[nn], 1+ Accumulate[ Prime[Range[nn]]^2]}],IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Sep 09 2014 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^2); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A122102 a(n) = Sum_{k=1..n} prime(k)^4.

Original entry on oeis.org

16, 97, 722, 3123, 17764, 46325, 129846, 260167, 540008, 1247289, 2170810, 4044971, 6870732, 10289533, 15169214, 23059695, 35177056, 49022897, 69174018, 94585699, 122983940, 161934021, 209392342, 272134583, 360663864, 464724265, 577275146, 708354747, 849512908

Offset: 1

Alexander Adamchuk, Aug 20 2006

nonn

a(n) is prime for n = {2,32,90,110,134,152,168,180,194,...} = A122127.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A007504, A024450, A098999, A122103, A122127.

Partial sums of A030514.

[&+[NthPrime(k)^4: k in [1..n]]: n in [1..30]]; // G. C. Greubel, Oct 02 2019

seq(add(ithprime(k)^4, k=1..n), n=1..30); # G. C. Greubel, Oct 02 2019

Table[Sum[Prime[k]^4,{k,1,n}],{n,1,100}]
Accumulate[Prime[Range[30]]^4] (* Harvey P. Dale, Aug 07 2021 *)

a(n)=my(s);forprime(p=2,prime(n),s+=p^4); s \\ Charles R Greathouse IV, Aug 02 2013

[sum(nth_prime(k)^4 for k in (1..n)) for n in (1..30)] # G. C. Greubel, Oct 02 2019

From Vladimir Shevelev, Aug 02 2013: (Start)
a(n) = 0.2*n^5*log(n)^4 + O(n^5*log(n)^3*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev).
A generalization: Sum_{i=1..n} prime(i)^k = 1/(k+1)*n^(k+1)*log(n)^k + O(n^(k+1)*log(n)^(k-1)*log(log(n))).
(End)

A128169 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^5 = 1 + A122103(k).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 22, 58, 155, 363, 464, 665, 1146, 2870, 3048, 4019, 5931, 8724, 21503, 50439, 67560, 476281, 705570, 4050684, 6956459, 7443590, 10449928, 10799546, 15385564, 17735139, 83325458, 245271750, 255583775, 1395860516, 2921734534, 6255577368, 9050771725, 12062893218, 13689205205, 42254229197, 46440930382

Offset: 1

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

a(52) > 3*10^13. - Bruce Garner, Jun 05 2021

a(53) > 1.2*10^14. - Bruce Garner, Mar 28 2022

Bruce Garner, Table of n, a(n) for n = 1..52
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.

p = 2; k = 0; s = 1; lst = {}; While[k < 521330000, s = s + p^5; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p]

a(31) from Sean A. Irvine, Jan 19 2011
a(32)-a(33) from Robert G. Wilson v, Jan 20 2011
a(34)-a(41) from Robert Price, Dec 18 2013

A098561 Numbers n such that the sum of the squares of the first n primes is prime.

Original entry on oeis.org

2, 18, 26, 36, 68, 78, 144, 158, 164, 174, 192, 212, 216, 236, 264, 288, 294, 338, 344, 356, 384, 404, 416, 426, 500, 516, 518, 522, 534, 540, 548, 614, 678, 680, 782, 858, 866, 876, 878, 896, 900, 912, 950, 974, 996, 1064, 1080, 1082, 1100, 1122, 1158, 1160

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

a(n) must clearly be even.

			2 is a term as the sum of the squares of the first two primes is 2^2 + 3^2 = 13, which is prime.

Zak Seidov, Table of n, a(n) for n = 1..1050

Cf. A098562 (corresponding primes), A024450 (sums of squares of primes), A098563 (sums of cubes of primes), A013916 (sums of primes).

Select[Range[1000], PrimeQ[Sum[Prime[i]^2, {i, #}]] &] (* Carl Najafi, Aug 22 2011 *)

A098562 Primes that are the sum of the squares of the first k primes for some k.

Original entry on oeis.org

13, 20477, 75997, 239087, 2210983, 3579761, 29194283, 40002073, 45448471, 55600481, 77290091, 108095623, 114986483, 155637463, 226226771, 302920139, 324657881, 519681709, 551321299, 618359839, 797005427, 944007487, 1039681147, 1124764853, 1923614047, 2135308631

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

These are the primes arising in A098561.

			From _K. D. Bajpai_, Dec 15 2014: (Start)
13 is in the sequence because the sum of the squares of the first 2 primes is 2^2 + 3^2 = 4 + 9 = 13, which is prime.
20477 is in the sequence because the sum of the squares of the first 18 primes is 2^2 + 3^2 + 5^2 + ... + 59^2 + 61^2 = 4 + 9 + 25 + ... + 3481 + 3721 = 20477, which is prime.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..15000

Cf. A098561 (corresponding n), A024450 (sum of squares of primes), A066525 (sums of cubes of primes), A013918 (sums of primes).

Cf. A000040, A006567. - Jonathan Vos Post, Aug 13 2009

Select[Table[Sum[Prime[k]^2, {k, 1, n}], {n, 1000}], PrimeQ]  (* K. D. Bajpai, Dec 15 2014 *)

s=0; forprime(p=2, 1e6, t=s+=p^2; if(isprime(t), print1(t,", "))) \\ K. D. Bajpai, Dec 15 2014

a(24)-a(26) from K. D. Bajpai, Dec 15 2014

a(42) in b-file corrected by Andrew Howroyd, Feb 28 2018

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

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

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A217601 Integer averages of squares of first primes.

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A233523 Prime(n), where n is such that (1+sum_{i=1..n} prime(i)) / n is an integer.

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A233862 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^2) / k is an integer.

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A122102 a(n) = Sum_{k=1..n} prime(k)^4.

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