A111441 -id:A111441 - cp's OEIS frontend

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

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

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

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

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

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

cp's OEIS Frontend

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

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A050248 Integer averages of first k primes for some 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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A085450 a(n) is the smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.

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