A217600 -id:A217600 - cp's OEIS frontend

A050248 Integer averages of first k primes for some k.

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

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

Original entry on oeis.org

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

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

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

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

cp's OEIS Frontend

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

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

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

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